/*
 * Copyright (c) Meta Platforms, Inc. and affiliates.
 * All rights reserved.
 *
 * This source code is licensed under the BSD-style license found in the
 * LICENSE file in the root directory of this source tree.
 */

#pragma once

#include <ATen/NumericUtils.h>
#include <c10/macros/Macros.h>
#include <c10/util/MathConstants.h>
#include <c10/util/complex.h>
#include "AccumulateType.h"
#include "General.h"
#include "Numerics.h"

namespace at {
namespace AtenIpexTypeXPU {

template <typename scalar_t>
static inline scalar_t zeta(scalar_t _x, scalar_t _q) {
  using acc_t = acc_type<scalar_t>;
  const acc_t MACHEP = acc_t{1.11022302462515654042E-16};
  constexpr acc_t zero = acc_t{0.0};
  constexpr acc_t half = acc_t{0.5};
  constexpr acc_t one = acc_t{1.0};
  static const acc_t A[] = {
      12.0,
      -720.0,
      30240.0,
      -1209600.0,
      47900160.0,
      -1.8924375803183791606e9, /*1.307674368e12/691*/
      7.47242496e10,
      -2.950130727918164224e12, /*1.067062284288e16/3617*/
      1.1646782814350067249e14, /*5.109094217170944e18/43867*/
      -4.5979787224074726105e15, /*8.028576626982912e20/174611*/
      1.8152105401943546773e17, /*1.5511210043330985984e23/854513*/
      -7.1661652561756670113e18 /*1.6938241367317436694528e27/236364091*/
  };
  acc_t x = static_cast<acc_t>(_x);
  acc_t q = static_cast<acc_t>(_q);

  int i = 0;
  acc_t a, b, k, s, t, w;
  if (x == one) {
    return std::numeric_limits<scalar_t>::infinity();
  }

  if (x < one) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }

  if (q <= zero) {
    if (q == std::floor(q)) {
      return std::numeric_limits<scalar_t>::infinity();
    }
    if (x != std::floor(x)) {
      return std::numeric_limits<scalar_t>::quiet_NaN();
    }
  }

  s = std::pow(q, -x);
  a = q;
  i = 0;
  b = zero;
  while ((i < 9) || (a <= acc_t{9.0})) {
    i += 1;
    a += one;
    b = std::pow(a, -x);
    s += b;
    if ((-MACHEP * s < b) && (b < MACHEP * s)) {
      return static_cast<scalar_t>(s);
    }
  };

  w = a;
  s += b * w / (x - one);
  s -= half * b;
  a = one;
  k = zero;
  for (int i = 0; i < 12; i++) {
    a *= x + k;
    b /= w;
    t = a * b / A[i];
    s = s + t;
    t = std::fabs(t / s);
    if (t < MACHEP) {
      return static_cast<scalar_t>(s);
    }
    k += one;
    a *= x + k;
    b /= w;
    k += one;
  }
  return static_cast<scalar_t>(s);
}

template <typename scalar_t>
static scalar_t ratevl(
    scalar_t x,
    const scalar_t num[],
    int64_t M,
    const scalar_t denom[],
    int64_t N) {
  // evaluating rational function, i.e., the ratio of two polynomials
  // the coefficients for numerator are given by `num` while coeffs for
  // denumerator are given by `denom`
  using acc_t = acc_type<scalar_t>;
  int64_t i, dir;
  scalar_t y, num_ans, denom_ans;
  acc_t x_ = static_cast<acc_t>(x);
  scalar_t absx = std::fabs(x_);
  const scalar_t* p;

  if (absx > 1) {
    /* Evaluate as a polynomial in 1/x. */
    dir = -1;
    p = num + M;
    y = 1 / x;
  } else {
    dir = 1;
    p = num;
    y = x;
  }

  /* Evaluate the numerator */
  num_ans = *p;
  p += dir;
  for (i = 1; i <= M; i++) {
    num_ans = num_ans * y + *p;
    p += dir;
  }
  /* Evaluate the denominator */
  if (absx > 1) {
    p = denom + N;
  } else {
    p = denom;
  }

  denom_ans = *p;
  p += dir;
  for (i = 1; i <= N; i++) {
    denom_ans = denom_ans * y + *p;
    p += dir;
  }
  if (absx > 1) {
    i = N - M;
    return Numerics<scalar_t>::pow(x, i) * num_ans / denom_ans;
  } else {
    return num_ans / denom_ans;
  }
}

// SciPy's lanczos implementation is taken from Boost
/* (C) Copyright John Maddock 2006.
 * Use, modification and distribution are subject to the
 * Boost Software License, Version 1.0. See
 * https://www.boost.org/LICENSE_1_0.txt or see NOTICE.
 */
template <typename scalar_t>
static scalar_t lanczos_sum_expg_scaled(scalar_t x) {
  // lanczos approximation
  scalar_t lanczos_sum_expg_scaled_num[13] = {
      0.006061842346248906525783753964555936883222,
      0.5098416655656676188125178644804694509993,
      19.51992788247617482847860966235652136208,
      449.9445569063168119446858607650988409623,
      6955.999602515376140356310115515198987526,
      75999.29304014542649875303443598909137092,
      601859.6171681098786670226533699352302507,
      3481712.15498064590882071018964774556468,
      14605578.08768506808414169982791359218571,
      43338889.32467613834773723740590533316085,
      86363131.28813859145546927288977868422342,
      103794043.1163445451906271053616070238554,
      56906521.91347156388090791033559122686859};
  scalar_t lanczos_sum_expg_scaled_denom[13] = {
      1.,
      66.,
      1925.,
      32670.,
      357423.,
      2637558.,
      13339535.,
      45995730.,
      105258076.,
      150917976.,
      120543840.,
      39916800.,
      0.};
  return ratevl(
      x,
      lanczos_sum_expg_scaled_num,
      sizeof(lanczos_sum_expg_scaled_num) /
              sizeof(lanczos_sum_expg_scaled_num[0]) -
          1,
      lanczos_sum_expg_scaled_denom,
      sizeof(lanczos_sum_expg_scaled_denom) /
              sizeof(lanczos_sum_expg_scaled_denom[0]) -
          1);
}

template <typename scalar_t>
static inline scalar_t bessel_j0_forward(scalar_t x) {
  const scalar_t PP[] = {
      +7.96936729297347051624e-04,
      +8.28352392107440799803e-02,
      +1.23953371646414299388e+00,
      +5.44725003058768775090e+00,
      +8.74716500199817011941e+00,
      +5.30324038235394892183e+00,
      +9.99999999999999997821e-01,
  };

  const scalar_t PQ[] = {
      +9.24408810558863637013e-04,
      +8.56288474354474431428e-02,
      +1.25352743901058953537e+00,
      +5.47097740330417105182e+00,
      +8.76190883237069594232e+00,
      +5.30605288235394617618e+00,
      +1.00000000000000000218e+00,
  };

  const scalar_t QP[] = {
      -1.13663838898469149931e-02,
      -1.28252718670509318512e+00,
      -1.95539544257735972385e+01,
      -9.32060152123768231369e+01,
      -1.77681167980488050595e+02,
      -1.47077505154951170175e+02,
      -5.14105326766599330220e+01,
      -6.05014350600728481186e+00,
  };

  const scalar_t QQ[] = {
      +6.43178256118178023184e+01,
      +8.56430025976980587198e+02,
      +3.88240183605401609683e+03,
      +7.24046774195652478189e+03,
      +5.93072701187316984827e+03,
      +2.06209331660327847417e+03,
      +2.42005740240291393179e+02,
  };

  const scalar_t RP[] = {
      -4.79443220978201773821e+09,
      +1.95617491946556577543e+12,
      -2.49248344360967716204e+14,
      +9.70862251047306323952e+15,
  };

  const scalar_t RQ[] = {
      +4.99563147152651017219e+02,
      +1.73785401676374683123e+05,
      +4.84409658339962045305e+07,
      +1.11855537045356834862e+10,
      +2.11277520115489217587e+12,
      +3.10518229857422583814e+14,
      +3.18121955943204943306e+16,
      +1.71086294081043136091e+18,
  };

  if (x < scalar_t(0)) {
    x = -x;
  }

  if (x <= scalar_t(5.0)) {
    if (x < scalar_t(0.00001)) {
      return scalar_t(1.0) - x * x / scalar_t(4.0);
    }

    scalar_t rp = 0.0;

    for (uint8_t index = 0; index <= 3; index++) {
      rp = rp * (x * x) + RP[index];
    }

    scalar_t rq = 0.0;

    for (uint8_t index = 0; index <= 7; index++) {
      rq = rq * (x * x) + RQ[index];
    }

    return (x * x - scalar_t(5.78318596294678452118e+00)) *
        (x * x - scalar_t(3.04712623436620863991e+01)) * rp / rq;
  }

  scalar_t pp = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pp = pp * (scalar_t(25.0) / (x * x)) + PP[index];
  }

  scalar_t pq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pq = pq * (scalar_t(25.0) / (x * x)) + PQ[index];
  }

  scalar_t qp = 0.0;

  for (uint8_t index = 0; index <= 7; index++) {
    qp = qp * (scalar_t(25.0) / (x * x)) + QP[index];
  }

  scalar_t qq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    qq = qq * (scalar_t(25.0) / (x * x)) + QQ[index];
  }

  return (pp / pq *
              Numerics<scalar_t>::cos(
                  x - scalar_t(0.785398163397448309615660845819875721)) -
          scalar_t(5.0) / x * (qp / qq) *
              Numerics<scalar_t>::sin(
                  x - scalar_t(0.785398163397448309615660845819875721))) *
      scalar_t(0.797884560802865355879892119868763737) /
      Numerics<scalar_t>::sqrt(x);
} // bessel_j0_forward(scalar_t x)

template <typename scalar_t>
static inline scalar_t bessel_j1_forward(scalar_t x) {
  const scalar_t PP[] = {
      +7.62125616208173112003e-04,
      +7.31397056940917570436e-02,
      +1.12719608129684925192e+00,
      +5.11207951146807644818e+00,
      +8.42404590141772420927e+00,
      +5.21451598682361504063e+00,
      +1.00000000000000000254e+00,
  };

  const scalar_t PQ[] = {
      +5.71323128072548699714e-04,
      +6.88455908754495404082e-02,
      +1.10514232634061696926e+00,
      +5.07386386128601488557e+00,
      +8.39985554327604159757e+00,
      +5.20982848682361821619e+00,
      +9.99999999999999997461e-01,
  };

  const scalar_t QP[] = {
      +5.10862594750176621635e-02,
      +4.98213872951233449420e+00,
      +7.58238284132545283818e+01,
      +3.66779609360150777800e+02,
      +7.10856304998926107277e+02,
      +5.97489612400613639965e+02,
      +2.11688757100572135698e+02,
      +2.52070205858023719784e+01,
  };

  const scalar_t QQ[] = {
      +7.42373277035675149943e+01,
      +1.05644886038262816351e+03,
      +4.98641058337653607651e+03,
      +9.56231892404756170795e+03,
      +7.99704160447350683650e+03,
      +2.82619278517639096600e+03,
      +3.36093607810698293419e+02,
  };

  const scalar_t RP[] = {
      -8.99971225705559398224e+08,
      +4.52228297998194034323e+11,
      -7.27494245221818276015e+13,
      +3.68295732863852883286e+15,
  };

  const scalar_t RQ[] = {
      +6.20836478118054335476e+02,
      +2.56987256757748830383e+05,
      +8.35146791431949253037e+07,
      +2.21511595479792499675e+10,
      +4.74914122079991414898e+12,
      +7.84369607876235854894e+14,
      +8.95222336184627338078e+16,
      +5.32278620332680085395e+18,
  };

  if (x <= scalar_t(5.0)) {
    scalar_t rp = 0.0;

    for (uint8_t index = 0; index <= 3; index++) {
      rp = rp * (x * x) + RP[index];
    }

    scalar_t rq = 0.0;

    for (uint8_t index = 0; index <= 7; index++) {
      rq = rq * (x * x) + RQ[index];
    }

    return rp / rq * x * (x * x - scalar_t(1.46819706421238932572e+01)) *
        (x * x - scalar_t(4.92184563216946036703e+01));
  }

  scalar_t pp = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pp = pp * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + PP[index];
  }

  scalar_t pq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pq = pq * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + PQ[index];
  }

  scalar_t qp = 0.0;

  for (uint8_t index = 0; index <= 7; index++) {
    qp = qp * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + QP[index];
  }

  scalar_t qq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    qq = qq * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + QQ[index];
  }

  return (pp / pq *
              Numerics<scalar_t>::cos(
                  x - scalar_t(2.356194490192344928846982537459627163)) -
          scalar_t(5.0) / x * (qp / qq) *
              Numerics<scalar_t>::sin(
                  x - scalar_t(2.356194490192344928846982537459627163))) *
      scalar_t(0.797884560802865355879892119868763737) /
      Numerics<scalar_t>::sqrt(x);
} // bessel_j1_forward(scalar_t x)

template <typename scalar_t>
static inline scalar_t bessel_y0_forward(scalar_t x) {
  const scalar_t PP[] = {
      +7.96936729297347051624e-04,
      +8.28352392107440799803e-02,
      +1.23953371646414299388e+00,
      +5.44725003058768775090e+00,
      +8.74716500199817011941e+00,
      +5.30324038235394892183e+00,
      +9.99999999999999997821e-01,
  };

  const scalar_t PQ[] = {
      +9.24408810558863637013e-04,
      +8.56288474354474431428e-02,
      +1.25352743901058953537e+00,
      +5.47097740330417105182e+00,
      +8.76190883237069594232e+00,
      +5.30605288235394617618e+00,
      +1.00000000000000000218e+00,
  };

  const scalar_t QP[] = {
      -1.13663838898469149931e-02,
      -1.28252718670509318512e+00,
      -1.95539544257735972385e+01,
      -9.32060152123768231369e+01,
      -1.77681167980488050595e+02,
      -1.47077505154951170175e+02,
      -5.14105326766599330220e+01,
      -6.05014350600728481186e+00,
  };

  const scalar_t QQ[] = {
      +6.43178256118178023184e+01,
      +8.56430025976980587198e+02,
      +3.88240183605401609683e+03,
      +7.24046774195652478189e+03,
      +5.93072701187316984827e+03,
      +2.06209331660327847417e+03,
      +2.42005740240291393179e+02,
  };

  const scalar_t YP[] = {
      +1.55924367855235737965e+04,
      -1.46639295903971606143e+07,
      +5.43526477051876500413e+09,
      -9.82136065717911466409e+11,
      +8.75906394395366999549e+13,
      -3.46628303384729719441e+15,
      +4.42733268572569800351e+16,
      -1.84950800436986690637e+16,
  };

  const scalar_t YQ[] = {
      +1.04128353664259848412e+03,
      +6.26107330137134956842e+05,
      +2.68919633393814121987e+08,
      +8.64002487103935000337e+10,
      +2.02979612750105546709e+13,
      +3.17157752842975028269e+15,
      +2.50596256172653059228e+17,
  };

  if (x <= scalar_t(5.0)) {
    if (x == scalar_t(0.0)) {
      return -std::numeric_limits<scalar_t>::infinity();
    }

    if (x < scalar_t(0.0)) {
      return std::numeric_limits<scalar_t>::quiet_NaN();
    }

    scalar_t yp = 0.0;

    for (uint8_t index = 0; index <= 7; index++) {
      yp = yp * (x * x) + YP[index];
    }

    scalar_t yq = 0.0;

    for (uint8_t index = 0; index <= 6; index++) {
      yq = yq * (x * x) + YQ[index];
    }

    return yp / yq +
        (scalar_t(0.636619772367581343075535053490057448) *
         Numerics<scalar_t>::log(x) * bessel_j0_forward(x));
  }

  scalar_t pp = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pp = pp * (scalar_t(25.0) / (x * x)) + PP[index];
  }

  scalar_t pq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pq = pq * (scalar_t(25.0) / (x * x)) + PQ[index];
  }

  scalar_t qp = 0.0;

  for (uint8_t index = 0; index <= 7; index++) {
    qp = qp * (scalar_t(25.0) / (x * x)) + QP[index];
  }

  scalar_t qq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    qq = qq * (scalar_t(25.0) / (x * x)) + QQ[index];
  }

  return (pp / pq *
              Numerics<scalar_t>::sin(
                  x - scalar_t(0.785398163397448309615660845819875721)) +
          scalar_t(5.0) / x * (qp / qq) *
              Numerics<scalar_t>::cos(
                  x - scalar_t(0.785398163397448309615660845819875721))) *
      scalar_t(0.797884560802865355879892119868763737) /
      Numerics<scalar_t>::sqrt(x);
} // bessel_y0_forward(scalar_t x)

template <typename scalar_t>
static inline scalar_t bessel_y1_forward(scalar_t x) {
  const scalar_t PP[] = {
      +7.62125616208173112003e-04,
      +7.31397056940917570436e-02,
      +1.12719608129684925192e+00,
      +5.11207951146807644818e+00,
      +8.42404590141772420927e+00,
      +5.21451598682361504063e+00,
      +1.00000000000000000254e+00,
  };

  const scalar_t PQ[] = {
      +5.71323128072548699714e-04,
      +6.88455908754495404082e-02,
      +1.10514232634061696926e+00,
      +5.07386386128601488557e+00,
      +8.39985554327604159757e+00,
      +5.20982848682361821619e+00,
      +9.99999999999999997461e-01,
  };

  const scalar_t QP[] = {
      +5.10862594750176621635e-02,
      +4.98213872951233449420e+00,
      +7.58238284132545283818e+01,
      +3.66779609360150777800e+02,
      +7.10856304998926107277e+02,
      +5.97489612400613639965e+02,
      +2.11688757100572135698e+02,
      +2.52070205858023719784e+01,
  };

  const scalar_t QQ[] = {
      +7.42373277035675149943e+01,
      +1.05644886038262816351e+03,
      +4.98641058337653607651e+03,
      +9.56231892404756170795e+03,
      +7.99704160447350683650e+03,
      +2.82619278517639096600e+03,
      +3.36093607810698293419e+02,
  };

  const scalar_t YP[] = {
      +1.26320474790178026440e+09,
      -6.47355876379160291031e+11,
      +1.14509511541823727583e+14,
      -8.12770255501325109621e+15,
      +2.02439475713594898196e+17,
      -7.78877196265950026825e+17,
  };

  const scalar_t YQ[] = {
      +5.94301592346128195359e+02,
      +2.35564092943068577943e+05,
      +7.34811944459721705660e+07,
      +1.87601316108706159478e+10,
      +3.88231277496238566008e+12,
      +6.20557727146953693363e+14,
      +6.87141087355300489866e+16,
      +3.97270608116560655612e+18,
  };

  if (x <= scalar_t(5.0)) {
    if (x == scalar_t(0.0)) {
      return -std::numeric_limits<scalar_t>::infinity();
    }

    if (x <= scalar_t(0.0)) {
      return std::numeric_limits<scalar_t>::quiet_NaN();
    }

    scalar_t yp = 0.0;

    for (uint8_t index = 0; index <= 5; index++) {
      yp = yp * (x * x) + YP[index];
    }

    scalar_t yq = 0.0;

    for (uint8_t index = 0; index <= 7; index++) {
      yq = yq * (x * x) + YQ[index];
    }

    return x * (yp / yq) +
        (scalar_t(0.636619772367581343075535053490057448) *
         (bessel_j1_forward(x) * Numerics<scalar_t>::log(x) -
          scalar_t(1.0) / x));
  }

  scalar_t pp = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pp = pp * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + PP[index];
  }

  scalar_t pq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    pq = pq * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + PQ[index];
  }

  scalar_t qp = 0.0;

  for (uint8_t index = 0; index <= 7; index++) {
    qp = qp * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + QP[index];
  }

  scalar_t qq = 0.0;

  for (uint8_t index = 0; index <= 6; index++) {
    qq = qq * (scalar_t(5.0) / x * (scalar_t(5.0) / x)) + QQ[index];
  }

  return (pp / pq *
              Numerics<scalar_t>::sin(
                  x - scalar_t(2.356194490192344928846982537459627163)) +
          scalar_t(5.0) / x * (qp / qq) *
              Numerics<scalar_t>::cos(
                  x - scalar_t(2.356194490192344928846982537459627163))) *
      scalar_t(0.797884560802865355879892119868763737) /
      Numerics<scalar_t>::sqrt(x);
} // bessel_y1_forward(scalar_t x)

template <typename scalar_t>
static inline scalar_t spherical_bessel_j0_forward(scalar_t x) {
  if (Numerics<scalar_t>::isinf(x)) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) < scalar_t(0.5)) {
    return scalar_t(1.0) +
        x * x *
        (scalar_t(-1.0) / scalar_t(6.0) +
         x * x *
             (scalar_t(1.0) / scalar_t(120.0) +
              x * x *
                  (scalar_t(-1.0) / scalar_t(5040.0) +
                   x * x *
                       (scalar_t(1.0) / scalar_t(362880.0) +
                        x * x *
                            (scalar_t(-1.0) / scalar_t(39916800.0) +
                             x * x *
                                 (scalar_t(1.0) / scalar_t(6227020800.0)))))));
  }

  return Numerics<scalar_t>::sin(x) / x;
} // spherical_bessel_j0_forward(T x)

template <typename scalar_t>
static inline scalar_t hermite_polynomial_he_forward(scalar_t x, scalar_t _n) {
  int n = static_cast<int64_t>(_n);
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x;
  scalar_t r;

  for (int64_t k = 1; k < n; k++) {
    r = x * q - k * p;
    p = q;
    q = r;
  }

  return r;
}

template <typename scalar_t>
static inline scalar_t hermite_polynomial_h_forward(scalar_t x, scalar_t _n) {
  int n = static_cast<int64_t>(_n);
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x + x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x + x;
  scalar_t r;

  for (int64_t k = 2; k < n + n; k += 2) {
    r = (x + x) * q - k * p;
    p = q;
    q = r;
  }

  return r;
}

template <typename scalar_t>
static inline scalar_t laguerre_polynomial_l_forward(scalar_t x, scalar_t _n) {
  int n = static_cast<int64_t>(_n);
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(0.0)) {
    return scalar_t(1.0);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return scalar_t(1.0) - x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = scalar_t(1.0) - x;
  scalar_t r;

  for (int64_t k = 1; k < n; k++) {
    r = (((k + k) + (scalar_t(1.0) - x)) * q - k * p) / (k + 1);
    p = q;
    q = r;
  }

  return r;
}

template <typename scalar_t>
static inline scalar_t legendre_polynomial_p_forward(scalar_t x, scalar_t _n) {
  int n = static_cast<int64_t>(_n);
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(1.0)) {
    if (x > scalar_t(0.0) || n % 2 == 0) {
      return scalar_t(1.0);
    }

    return scalar_t(-1.0);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x;
  scalar_t r;

  for (int64_t k = 1; k < n; k++) {
    r = ((k + k + 1) * x * q - k * p) / (k + 1);
    p = q;
    q = r;
  }

  return r;
}

template <typename scalar_t>
static inline scalar_t modified_bessel_i0_forward(scalar_t x) {
  const scalar_t A[] = {
      -4.41534164647933937950e-18, +3.33079451882223809783e-17,
      -2.43127984654795469359e-16, +1.71539128555513303061e-15,
      -1.16853328779934516808e-14, +7.67618549860493561688e-14,
      -4.85644678311192946090e-13, +2.95505266312963983461e-12,
      -1.72682629144155570723e-11, +9.67580903537323691224e-11,
      -5.18979560163526290666e-10, +2.65982372468238665035e-09,
      -1.30002500998624804212e-08, +6.04699502254191894932e-08,
      -2.67079385394061173391e-07, +1.11738753912010371815e-06,
      -4.41673835845875056359e-06, +1.64484480707288970893e-05,
      -5.75419501008210370398e-05, +1.88502885095841655729e-04,
      -5.76375574538582365885e-04, +1.63947561694133579842e-03,
      -4.32430999505057594430e-03, +1.05464603945949983183e-02,
      -2.37374148058994688156e-02, +4.93052842396707084878e-02,
      -9.49010970480476444210e-02, +1.71620901522208775349e-01,
      -3.04682672343198398683e-01, +6.76795274409476084995e-01,
  };

  const scalar_t B[] = {
      -7.23318048787475395456e-18, -4.83050448594418207126e-18,
      +4.46562142029675999901e-17, +3.46122286769746109310e-17,
      -2.82762398051658348494e-16, -3.42548561967721913462e-16,
      +1.77256013305652638360e-15, +3.81168066935262242075e-15,
      -9.55484669882830764870e-15, -4.15056934728722208663e-14,
      +1.54008621752140982691e-14, +3.85277838274214270114e-13,
      +7.18012445138366623367e-13, -1.79417853150680611778e-12,
      -1.32158118404477131188e-11, -3.14991652796324136454e-11,
      +1.18891471078464383424e-11, +4.94060238822496958910e-10,
      +3.39623202570838634515e-09, +2.26666899049817806459e-08,
      +2.04891858946906374183e-07, +2.89137052083475648297e-06,
      +6.88975834691682398426e-05, +3.36911647825569408990e-03,
      +8.04490411014108831608e-01,
  };

  scalar_t p;
  scalar_t q = 0.0;

  if (Numerics<scalar_t>::abs(x) <= scalar_t(8.0)) {
    scalar_t a = A[0];

    for (uint8_t index = 1; index < 30; index++) {
      p = q;
      q = a;
      a = ((Numerics<scalar_t>::abs(x) / scalar_t(2.0)) - scalar_t(2.0)) * q -
          p + A[index];
    }

    return Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x)) *
        (scalar_t(0.5) * (a - p));
  }

  scalar_t b = B[0];

  for (uint8_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(32.0) / Numerics<scalar_t>::abs(x) - scalar_t(2.0)) * q - p +
        B[index];
  }

  return Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x)) *
      (scalar_t(0.5) * (b - p)) /
      Numerics<scalar_t>::sqrt(Numerics<scalar_t>::abs(x));
} // modified_bessel_i0_forward(T x)

template <typename scalar_t>
static inline scalar_t modified_bessel_i1_forward(scalar_t x) {
  const scalar_t A[] = {
      +2.77791411276104639959e-18, -2.11142121435816608115e-17,
      +1.55363195773620046921e-16, -1.10559694773538630805e-15,
      +7.60068429473540693410e-15, -5.04218550472791168711e-14,
      +3.22379336594557470981e-13, -1.98397439776494371520e-12,
      +1.17361862988909016308e-11, -6.66348972350202774223e-11,
      +3.62559028155211703701e-10, -1.88724975172282928790e-09,
      +9.38153738649577178388e-09, -4.44505912879632808065e-08,
      +2.00329475355213526229e-07, -8.56872026469545474066e-07,
      +3.47025130813767847674e-06, -1.32731636560394358279e-05,
      +4.78156510755005422638e-05, -1.61760815825896745588e-04,
      +5.12285956168575772895e-04, -1.51357245063125314899e-03,
      +4.15642294431288815669e-03, -1.05640848946261981558e-02,
      +2.47264490306265168283e-02, -5.29459812080949914269e-02,
      +1.02643658689847095384e-01, -1.76416518357834055153e-01,
      +2.52587186443633654823e-01,
  };

  const scalar_t B[] = {
      +7.51729631084210481353e-18, +4.41434832307170791151e-18,
      -4.65030536848935832153e-17, -3.20952592199342395980e-17,
      +2.96262899764595013876e-16, +3.30820231092092828324e-16,
      -1.88035477551078244854e-15, -3.81440307243700780478e-15,
      +1.04202769841288027642e-14, +4.27244001671195135429e-14,
      -2.10154184277266431302e-14, -4.08355111109219731823e-13,
      -7.19855177624590851209e-13, +2.03562854414708950722e-12,
      +1.41258074366137813316e-11, +3.25260358301548823856e-11,
      -1.89749581235054123450e-11, -5.58974346219658380687e-10,
      -3.83538038596423702205e-09, -2.63146884688951950684e-08,
      -2.51223623787020892529e-07, -3.88256480887769039346e-06,
      -1.10588938762623716291e-04, -9.76109749136146840777e-03,
      +7.78576235018280120474e-01,
  };

  scalar_t p;
  scalar_t q = 0.0;

  if (Numerics<scalar_t>::abs(x) <= scalar_t(8.0)) {
    scalar_t a = A[0];

    for (uint8_t index = 1; index < 29; index++) {
      p = q;
      q = a;
      a = ((Numerics<scalar_t>::abs(x) / scalar_t(2.0)) - scalar_t(2.0)) * q -
          p + A[index];
    }

    if (x < scalar_t(0.0)) {
      return -(
          scalar_t(0.5) * (a - p) * Numerics<scalar_t>::abs(x) *
          Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x)));
    }

    return scalar_t(0.5) * (a - p) * Numerics<scalar_t>::abs(x) *
        Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x));
  }

  scalar_t b = B[0];

  for (uint8_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(32.0) / Numerics<scalar_t>::abs(x) - scalar_t(2.0)) * q - p +
        B[index];
  }

  if (x < scalar_t(0.0)) {
    return -(
        Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x)) *
        (scalar_t(0.5) * (b - p)) /
        Numerics<scalar_t>::sqrt(Numerics<scalar_t>::abs(x)));
  }

  return Numerics<scalar_t>::exp(Numerics<scalar_t>::abs(x)) *
      (scalar_t(0.5) * (b - p)) /
      Numerics<scalar_t>::sqrt(Numerics<scalar_t>::abs(x));
} // modified_bessel_i1_forward(T x)

template <typename scalar_t>
static inline scalar_t modified_bessel_k0_forward(scalar_t x) {
  const scalar_t A[] = {
      +1.37446543561352307156e-16,
      +4.25981614279661018399e-14,
      +1.03496952576338420167e-11,
      +1.90451637722020886025e-09,
      +2.53479107902614945675e-07,
      +2.28621210311945178607e-05,
      +1.26461541144692592338e-03,
      +3.59799365153615016266e-02,
      +3.44289899924628486886e-01,
      -5.35327393233902768720e-01,
  };

  const scalar_t B[] = {
      +5.30043377268626276149e-18, -1.64758043015242134646e-17,
      +5.21039150503902756861e-17, -1.67823109680541210385e-16,
      +5.51205597852431940784e-16, -1.84859337734377901440e-15,
      +6.34007647740507060557e-15, -2.22751332699166985548e-14,
      +8.03289077536357521100e-14, -2.98009692317273043925e-13,
      +1.14034058820847496303e-12, -4.51459788337394416547e-12,
      +1.85594911495471785253e-11, -7.95748924447710747776e-11,
      +3.57739728140030116597e-10, -1.69753450938905987466e-09,
      +8.57403401741422608519e-09, -4.66048989768794782956e-08,
      +2.76681363944501510342e-07, -1.83175552271911948767e-06,
      +1.39498137188764993662e-05, -1.28495495816278026384e-04,
      +1.56988388573005337491e-03, -3.14481013119645005427e-02,
      +2.44030308206595545468e+00,
  };

  if (x == scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::infinity();
  }

  if (x < scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }

  scalar_t p;
  scalar_t q = 0.0;

  if (x <= scalar_t(2.0)) {
    scalar_t a = A[0];

    for (uint8_t index = 1; index < 10; index++) {
      p = q;
      q = a;
      a = (x * x - scalar_t(2.0)) * q - p + A[index];
    }

    return scalar_t(0.5) * (a - p) -
        Numerics<scalar_t>::log(0.5 * x) * modified_bessel_i0_forward(x);
  }

  scalar_t b = B[0];

  for (uint8_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(8.0) / x - scalar_t(2.0)) * q - p + B[index];
  }

  return Numerics<scalar_t>::exp(-x) * (scalar_t(0.5) * (b - p)) /
      Numerics<scalar_t>::sqrt(x);
} // modified_bessel_k0_forward(T x)

template <typename scalar_t>
static inline scalar_t modified_bessel_k1_forward(scalar_t x) {
  const scalar_t A[] = {
      -7.02386347938628759343e-18,
      -2.42744985051936593393e-15,
      -6.66690169419932900609e-13,
      -1.41148839263352776110e-10,
      -2.21338763073472585583e-08,
      -2.43340614156596823496e-06,
      -1.73028895751305206302e-04,
      -6.97572385963986435018e-03,
      -1.22611180822657148235e-01,
      -3.53155960776544875667e-01,
      +1.52530022733894777053e+00,
  };

  const scalar_t B[] = {
      -5.75674448366501715755e-18, +1.79405087314755922667e-17,
      -5.68946255844285935196e-17, +1.83809354436663880070e-16,
      -6.05704724837331885336e-16, +2.03870316562433424052e-15,
      -7.01983709041831346144e-15, +2.47715442448130437068e-14,
      -8.97670518232499435011e-14, +3.34841966607842919884e-13,
      -1.28917396095102890680e-12, +5.13963967348173025100e-12,
      -2.12996783842756842877e-11, +9.21831518760500529508e-11,
      -4.19035475934189648750e-10, +2.01504975519703286596e-09,
      -1.03457624656780970260e-08, +5.74108412545004946722e-08,
      -3.50196060308781257119e-07, +2.40648494783721712015e-06,
      -1.93619797416608296024e-05, +1.95215518471351631108e-04,
      -2.85781685962277938680e-03, +1.03923736576817238437e-01,
      +2.72062619048444266945e+00,
  };

  if (x == scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::infinity();
  }

  if (x < scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }

  scalar_t p;
  scalar_t q = 0.0;

  if (x <= scalar_t(2.0)) {
    scalar_t a = A[0];

    for (uint8_t index = 1; index < 11; index++) {
      p = q;
      q = a;
      a = (x * x - scalar_t(2.0)) * q - p + A[index];
    }

    return Numerics<scalar_t>::log(scalar_t(0.5) * x) *
        modified_bessel_i1_forward(x) +
        scalar_t(0.5) * (a - p) / x;
  }

  scalar_t b = B[0];

  for (uint8_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(8.0) / x - scalar_t(2.0)) * q - p + B[index];
  }

  return Numerics<scalar_t>::exp(-x) * (scalar_t(0.5) * (b - p)) /
      Numerics<scalar_t>::sqrt(x);
} // modified_bessel_k1_forward(T x)

template <typename scalar_t>
static inline scalar_t scaled_modified_bessel_k0_forward(scalar_t x) {
  const scalar_t A[] = {
      +1.37446543561352307156e-16,
      +4.25981614279661018399e-14,
      +1.03496952576338420167e-11,
      +1.90451637722020886025e-09,
      +2.53479107902614945675e-07,
      +2.28621210311945178607e-05,
      +1.26461541144692592338e-03,
      +3.59799365153615016266e-02,
      +3.44289899924628486886e-01,
      -5.35327393233902768720e-01,
  };

  const scalar_t B[] = {
      +5.30043377268626276149e-18, -1.64758043015242134646e-17,
      +5.21039150503902756861e-17, -1.67823109680541210385e-16,
      +5.51205597852431940784e-16, -1.84859337734377901440e-15,
      +6.34007647740507060557e-15, -2.22751332699166985548e-14,
      +8.03289077536357521100e-14, -2.98009692317273043925e-13,
      +1.14034058820847496303e-12, -4.51459788337394416547e-12,
      +1.85594911495471785253e-11, -7.95748924447710747776e-11,
      +3.57739728140030116597e-10, -1.69753450938905987466e-09,
      +8.57403401741422608519e-09, -4.66048989768794782956e-08,
      +2.76681363944501510342e-07, -1.83175552271911948767e-06,
      +1.39498137188764993662e-05, -1.28495495816278026384e-04,
      +1.56988388573005337491e-03, -3.14481013119645005427e-02,
      +2.44030308206595545468e+00,
  };

  if (x == scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::infinity();
  }

  if (x < scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }

  scalar_t p;
  scalar_t q = 0.0;

  if (x <= scalar_t(2.0)) {
    scalar_t a = A[0];

    for (uint64_t index = 1; index < 10; index++) {
      p = q;
      q = a;
      a = (x * x - scalar_t(2.0)) * q - p + A[index];
    }

    return (scalar_t(0.5) * (a - p) -
            Numerics<scalar_t>::log(scalar_t(0.5) * x) *
                modified_bessel_i0_forward(x)) *
        Numerics<scalar_t>::exp(x);
  }

  scalar_t b = B[0];

  for (uint64_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(8.0) / x - scalar_t(2.0)) * q - p + B[index];
  }

  return scalar_t(0.5) * (b - p) / Numerics<scalar_t>::sqrt(x);
} // T scaled_modified_bessel_k0_forward(T x)

template <typename scalar_t>
static inline scalar_t scaled_modified_bessel_k1_forward(scalar_t x) {
  const scalar_t A[] = {
      -7.02386347938628759343e-18,
      -2.42744985051936593393e-15,
      -6.66690169419932900609e-13,
      -1.41148839263352776110e-10,
      -2.21338763073472585583e-08,
      -2.43340614156596823496e-06,
      -1.73028895751305206302e-04,
      -6.97572385963986435018e-03,
      -1.22611180822657148235e-01,
      -3.53155960776544875667e-01,
      +1.52530022733894777053e+00,
  };

  const scalar_t B[] = {
      -5.75674448366501715755e-18, +1.79405087314755922667e-17,
      -5.68946255844285935196e-17, +1.83809354436663880070e-16,
      -6.05704724837331885336e-16, +2.03870316562433424052e-15,
      -7.01983709041831346144e-15, +2.47715442448130437068e-14,
      -8.97670518232499435011e-14, +3.34841966607842919884e-13,
      -1.28917396095102890680e-12, +5.13963967348173025100e-12,
      -2.12996783842756842877e-11, +9.21831518760500529508e-11,
      -4.19035475934189648750e-10, +2.01504975519703286596e-09,
      -1.03457624656780970260e-08, +5.74108412545004946722e-08,
      -3.50196060308781257119e-07, +2.40648494783721712015e-06,
      -1.93619797416608296024e-05, +1.95215518471351631108e-04,
      -2.85781685962277938680e-03, +1.03923736576817238437e-01,
      +2.72062619048444266945e+00,
  };

  if (x == scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::infinity();
  }

  if (x < scalar_t(0.0)) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }

  scalar_t p;
  scalar_t q = 0.0;

  if (x <= scalar_t(2.0)) {
    scalar_t a = A[0];

    for (uint64_t index = 1; index < 11; index++) {
      p = q;
      q = a;
      a = (x * x - scalar_t(2.0)) * q - p + A[index];
    }

    return (Numerics<scalar_t>::log(scalar_t(0.5) * x) *
                modified_bessel_i1_forward(x) +
            scalar_t(0.5) * (a - p) / x) *
        Numerics<scalar_t>::exp(x);
  }

  scalar_t b = B[0];

  for (uint64_t index = 1; index < 25; index++) {
    p = q;
    q = b;
    b = (scalar_t(8.0) / x - scalar_t(2.0)) * q - p + B[index];
  }

  return (scalar_t(0.5) * (b - p) / Numerics<scalar_t>::sqrt(x));
} // T scaled_modified_bessel_k1_forward(T x)

template <typename scalar_t>
static scalar_t _igam_helper_fac(scalar_t a, scalar_t x) {
  // compute x^a * exp(-a) / gamma(a)
  // corrected from (15) and (16) in [igam2] by replacing exp(x - a) with
  // exp(a - x).
  using accscalar_t = acc_type<scalar_t>;

  accscalar_t ax, fac, res, num, numfac;
  accscalar_t MAXLOG = std::is_same<scalar_t, double>::value
      ? 7.09782712893383996843E2
      : 88.72283905206835;
  accscalar_t EXP1 = 2.718281828459045;
  accscalar_t lanczos_g = 6.024680040776729583740234375;

  accscalar_t a_x = static_cast<accscalar_t>(a - x);
  accscalar_t a_ = static_cast<accscalar_t>(a);
  if (std::fabs(a_x) > 0.4 * std::fabs(a_)) {
    ax = a * Numerics<scalar_t>::log(x) - x - Numerics<scalar_t>::lgamma(a_);
    if (ax < -MAXLOG) {
      return 0.0;
    }
    return Numerics<accscalar_t>::exp(ax);
  }

  fac = a + lanczos_g - 0.5;
  res = Numerics<accscalar_t>::sqrt(fac / EXP1) / lanczos_sum_expg_scaled(a);

  if ((a < 200) && (x < 200)) {
    res *= Numerics<accscalar_t>::exp(a - x) *
        Numerics<accscalar_t>::pow((x / fac), a);
  } else {
    num = x - a - lanczos_g + 0.5;
    numfac = num / fac;
    res *= Numerics<accscalar_t>::exp(
        a * (Numerics<accscalar_t>::log1p(numfac) - numfac) +
        x * (0.5 - lanczos_g) / fac);
  }
  return res;
}

template <typename scalar_t>
static inline scalar_t _igam_helper_series(scalar_t a, scalar_t x) {
  // Compute igam using DLMF 8.11.4. [igam1]

  using accscalar_t = acc_type<scalar_t>;
  static const accscalar_t MACHEP = std::is_same<accscalar_t, double>::value
      ? 1.11022302462515654042E-16
      : 5.9604644775390625E-8;
  static const int MAXITER = 2000;

  int i;
  accscalar_t ans, ax, c, r;

  ax = _igam_helper_fac(a, x);
  if (ax == 0.0) {
    return 0.0;
  }

  /* power series */
  r = a;
  c = 1.0;
  ans = 1.0;

  for (i = 0; i < MAXITER; i++) {
    r += 1.0;
    c *= x / r;
    ans += c;
    if (c <= MACHEP * ans) {
      break;
    }
  }
  return (ans * ax / a);
}

template <typename scalar_t>
static inline scalar_t _igamc_helper_series(scalar_t a, scalar_t x) {
  // Compute igamc using DLMF 8.7.3 [igam1]. This is related to the series in
  // _igam_helper_series but extra care is taken to avoid cancellation.

  using accscalar_t = acc_type<scalar_t>;
  int n;
  accscalar_t fac = 1;
  accscalar_t sum = 0;
  accscalar_t term, logx;
  static const int MAXITER = 2000;
  static const accscalar_t MACHEP = std::is_same<accscalar_t, double>::value
      ? 1.11022302462515654042E-16
      : 5.9604644775390625E-8;

  for (n = 1; n < MAXITER; n++) {
    fac *= -x / n;
    term = fac / (a + n);
    sum += term;
    if (std::fabs(term) <= MACHEP * std::fabs(sum)) {
      break;
    }
  }

  logx = Numerics<scalar_t>::log(x);
  term =
      -Numerics<scalar_t>::expm1(a * logx - Numerics<scalar_t>::lgamma(1 + a));
  return term -
      Numerics<scalar_t>::exp(a * logx - Numerics<scalar_t>::lgamma(a)) * sum;
}

template <typename scalar_t>
static inline scalar_t _igam_helper_asymptotic_series(
    scalar_t a,
    scalar_t x,
    bool igam) {
  // Compute igam/igamc using DLMF 8.12.3/8.12.4 [igam1]

  using accscalar_t = acc_type<scalar_t>;
  static const accscalar_t d[25][25] = {
      {-3.3333333333333333e-1,  8.3333333333333333e-2,
       -1.4814814814814815e-2,  1.1574074074074074e-3,
       3.527336860670194e-4,    -1.7875514403292181e-4,
       3.9192631785224378e-5,   -2.1854485106799922e-6,
       -1.85406221071516e-6,    8.296711340953086e-7,
       -1.7665952736826079e-7,  6.7078535434014986e-9,
       1.0261809784240308e-8,   -4.3820360184533532e-9,
       9.1476995822367902e-10,  -2.551419399494625e-11,
       -5.8307721325504251e-11, 2.4361948020667416e-11,
       -5.0276692801141756e-12, 1.1004392031956135e-13,
       3.3717632624009854e-13,  -1.3923887224181621e-13,
       2.8534893807047443e-14,  -5.1391118342425726e-16,
       -1.9752288294349443e-15},
      {-1.8518518518518519e-3,  -3.4722222222222222e-3,  2.6455026455026455e-3,
       -9.9022633744855967e-4,  2.0576131687242798e-4,   -4.0187757201646091e-7,
       -1.8098550334489978e-5,  7.6491609160811101e-6,   -1.6120900894563446e-6,
       4.6471278028074343e-9,   1.378633446915721e-7,    -5.752545603517705e-8,
       1.1951628599778147e-8,   -1.7543241719747648e-11, -1.0091543710600413e-9,
       4.1627929918425826e-10,  -8.5639070264929806e-11, 6.0672151016047586e-14,
       7.1624989648114854e-12,  -2.9331866437714371e-12, 5.9966963656836887e-13,
       -2.1671786527323314e-16, -4.9783399723692616e-14, 2.0291628823713425e-14,
       -4.13125571381061e-15},
      {4.1335978835978836e-3,  -2.6813271604938272e-3,  7.7160493827160494e-4,
       2.0093878600823045e-6,  -1.0736653226365161e-4,  5.2923448829120125e-5,
       -1.2760635188618728e-5, 3.4235787340961381e-8,   1.3721957309062933e-6,
       -6.298992138380055e-7,  1.4280614206064242e-7,   -2.0477098421990866e-10,
       -1.4092529910867521e-8, 6.228974084922022e-9,    -1.3670488396617113e-9,
       9.4283561590146782e-13, 1.2872252400089318e-10,  -5.5645956134363321e-11,
       1.1975935546366981e-11, -4.1689782251838635e-15, -1.0940640427884594e-12,
       4.6622399463901357e-13, -9.905105763906906e-14,  1.8931876768373515e-17,
       8.8592218725911273e-15},
      {6.4943415637860082e-4,   2.2947209362139918e-4,  -4.6918949439525571e-4,
       2.6772063206283885e-4,   -7.5618016718839764e-5, -2.3965051138672967e-7,
       1.1082654115347302e-5,   -5.6749528269915966e-6, 1.4230900732435884e-6,
       -2.7861080291528142e-11, -1.6958404091930277e-7, 8.0994649053880824e-8,
       -1.9111168485973654e-8,  2.3928620439808118e-12, 2.0620131815488798e-9,
       -9.4604966618551322e-10, 2.1541049775774908e-10, -1.388823336813903e-14,
       -2.1894761681963939e-11, 9.7909989511716851e-12, -2.1782191880180962e-12,
       6.2088195734079014e-17,  2.126978363279737e-13,  -9.3446887915174333e-14,
       2.0453671226782849e-14},
      {-8.618882909167117e-4,   7.8403922172006663e-4,
       -2.9907248030319018e-4,  -1.4638452578843418e-6,
       6.6414982154651222e-5,   -3.9683650471794347e-5,
       1.1375726970678419e-5,   2.5074972262375328e-10,
       -1.6954149536558306e-6,  8.9075075322053097e-7,
       -2.2929348340008049e-7,  2.956794137544049e-11,
       2.8865829742708784e-8,   -1.4189739437803219e-8,
       3.4463580499464897e-9,   -2.3024517174528067e-13,
       -3.9409233028046405e-10, 1.8602338968504502e-10,
       -4.356323005056618e-11,  1.2786001016296231e-15,
       4.6792750266579195e-12,  -2.1492464706134829e-12,
       4.9088156148096522e-13,  -6.3385914848915603e-18,
       -5.0453320690800944e-14},
      {-3.3679855336635815e-4, -6.9728137583658578e-5,  2.7727532449593921e-4,
       -1.9932570516188848e-4, 6.7977804779372078e-5,   1.419062920643967e-7,
       -1.3594048189768693e-5, 8.0184702563342015e-6,   -2.2914811765080952e-6,
       -3.252473551298454e-10, 3.4652846491085265e-7,   -1.8447187191171343e-7,
       4.8240967037894181e-8,  -1.7989466721743515e-14, -6.3061945000135234e-9,
       3.1624176287745679e-9,  -7.8409242536974293e-10, 5.1926791652540407e-15,
       9.3589442423067836e-11, -4.5134262161632782e-11, 1.0799129993116827e-11,
       -3.661886712685252e-17, -1.210902069055155e-12,  5.6807435849905643e-13,
       -1.3249659916340829e-13},
      {5.3130793646399222e-4,  -5.9216643735369388e-4,  2.7087820967180448e-4,
       7.9023532326603279e-7,  -8.1539693675619688e-5,  5.6116827531062497e-5,
       -1.8329116582843376e-5, -3.0796134506033048e-9,  3.4651553688036091e-6,
       -2.0291327396058604e-6, 5.7887928631490037e-7,   2.338630673826657e-13,
       -8.8286007463304835e-8, 4.7435958880408128e-8,   -1.2545415020710382e-8,
       8.6496488580102925e-14, 1.6846058979264063e-9,   -8.5754928235775947e-10,
       2.1598224929232125e-10, -7.6132305204761539e-16, -2.6639822008536144e-11,
       1.3065700536611057e-11, -3.1799163902367977e-12, 4.7109761213674315e-18,
       3.6902800842763467e-13},
      {3.4436760689237767e-4,   5.1717909082605922e-5,
       -3.3493161081142236e-4,  2.812695154763237e-4,
       -1.0976582244684731e-4,  -1.2741009095484485e-7,
       2.7744451511563644e-5,   -1.8263488805711333e-5,
       5.7876949497350524e-6,   4.9387589339362704e-10,
       -1.0595367014026043e-6,  6.1667143761104075e-7,
       -1.7562973359060462e-7,  -1.2974473287015439e-12,
       2.695423606288966e-8,    -1.4578352908731271e-8,
       3.887645959386175e-9,    -3.8810022510194121e-17,
       -5.3279941738772867e-10, 2.7437977643314845e-10,
       -6.9957960920705679e-11, 2.5899863874868481e-17,
       8.8566890996696381e-12,  -4.403168815871311e-12,
       1.0865561947091654e-12},
      {-6.5262391859530942e-4,  8.3949872067208728e-4,  -4.3829709854172101e-4,
       -6.969091458420552e-7,   1.6644846642067548e-4,  -1.2783517679769219e-4,
       4.6299532636913043e-5,   4.5579098679227077e-9,  -1.0595271125805195e-5,
       6.7833429048651666e-6,   -2.1075476666258804e-6, -1.7213731432817145e-11,
       3.7735877416110979e-7,   -2.1867506700122867e-7, 6.2202288040189269e-8,
       6.5977038267330006e-16,  -9.5903864974256858e-9, 5.2132144922808078e-9,
       -1.3991589583935709e-9,  5.382058999060575e-16,  1.9484714275467745e-10,
       -1.0127287556389682e-10, 2.6077347197254926e-11, -5.0904186999932993e-18,
       -3.3721464474854592e-12},
      {-5.9676129019274625e-4,  -7.2048954160200106e-5,
       6.7823088376673284e-4,   -6.4014752602627585e-4,
       2.7750107634328704e-4,   1.8197008380465151e-7,
       -8.4795071170685032e-5,  6.105192082501531e-5,
       -2.1073920183404862e-5,  -8.8585890141255994e-10,
       4.5284535953805377e-6,   -2.8427815022504408e-6,
       8.7082341778646412e-7,   3.6886101871706965e-12,
       -1.5344695190702061e-7,  8.862466778790695e-8,
       -2.5184812301826817e-8,  -1.0225912098215092e-14,
       3.8969470758154777e-9,   -2.1267304792235635e-9,
       5.7370135528051385e-10,  -1.887749850169741e-19,
       -8.0931538694657866e-11, 4.2382723283449199e-11,
       -1.1002224534207726e-11},
      {1.3324454494800656e-3,   -1.9144384985654775e-3,  1.1089369134596637e-3,
       9.932404122642299e-7,    -5.0874501293093199e-4,  4.2735056665392884e-4,
       -1.6858853767910799e-4,  -8.1301893922784998e-9,  4.5284402370562147e-5,
       -3.127053674781734e-5,   1.044986828530338e-5,    4.8435226265680926e-11,
       -2.1482565873456258e-6,  1.329369701097492e-6,    -4.0295693092101029e-7,
       -1.7567877666323291e-13, 7.0145043163668257e-8,   -4.040787734999483e-8,
       1.1474026743371963e-8,   3.9642746853563325e-18,  -1.7804938269892714e-9,
       9.7480262548731646e-10,  -2.6405338676507616e-10, 5.794875163403742e-18,
       3.7647749553543836e-11},
      {1.579727660730835e-3,   1.6251626278391582e-4,   -2.0633421035543276e-3,
       2.1389686185689098e-3,  -1.0108559391263003e-3,  -3.9912705529919201e-7,
       3.6235025084764691e-4,  -2.8143901463712154e-4,  1.0449513336495887e-4,
       2.1211418491830297e-9,  -2.5779417251947842e-5,  1.7281818956040463e-5,
       -5.6413773872904282e-6, -1.1024320105776174e-11, 1.1223224418895175e-6,
       -6.8693396379526735e-7, 2.0653236975414887e-7,   4.6714772409838506e-14,
       -3.5609886164949055e-8, 2.0470855345905963e-8,   -5.8091738633283358e-9,
       -1.332821287582869e-16, 9.0354604391335133e-10,  -4.9598782517330834e-10,
       1.3481607129399749e-10},
      {-4.0725121195140166e-3, 6.4033628338080698e-3,  -4.0410161081676618e-3,
       -2.183732802866233e-6,  2.1740441801254639e-3,  -1.9700440518418892e-3,
       8.3595469747962458e-4,  1.9445447567109655e-8,  -2.5779387120421696e-4,
       1.9009987368139304e-4,  -6.7696499937438965e-5, -1.4440629666426572e-10,
       1.5712512518742269e-5,  -1.0304008744776893e-5, 3.304517767401387e-6,
       7.9829760242325709e-13, -6.4097794149313004e-7, 3.8894624761300056e-7,
       -1.1618347644948869e-7, -2.816808630596451e-15, 1.9878012911297093e-8,
       -1.1407719956357511e-8, 3.2355857064185555e-9,  4.1759468293455945e-20,
       -5.0423112718105824e-10},
      {-5.9475779383993003e-3, -5.4016476789260452e-4, 8.7910413550767898e-3,
       -9.8576315587856125e-3, 5.0134695031021538e-3,  1.2807521786221875e-6,
       -2.0626019342754683e-3, 1.7109128573523058e-3,  -6.7695312714133799e-4,
       -6.9011545676562133e-9, 1.8855128143995902e-4,  -1.3395215663491969e-4,
       4.6263183033528039e-5,  4.0034230613321351e-11, -1.0255652921494033e-5,
       6.612086372797651e-6,   -2.0913022027253008e-6, -2.0951775649603837e-13,
       3.9756029041993247e-7,  -2.3956211978815887e-7, 7.1182883382145864e-8,
       8.925574873053455e-16,  -1.2101547235064676e-8, 6.9350618248334386e-9,
       -1.9661464453856102e-9},
      {1.7402027787522711e-2,   -2.9527880945699121e-2, 2.0045875571402799e-2,
       7.0289515966903407e-6,   -1.2375421071343148e-2, 1.1976293444235254e-2,
       -5.4156038466518525e-3,  -6.3290893396418616e-8, 1.8855118129005065e-3,
       -1.473473274825001e-3,   5.5515810097708387e-4,  5.2406834412550662e-10,
       -1.4357913535784836e-4,  9.9181293224943297e-5,  -3.3460834749478311e-5,
       -3.5755837291098993e-12, 7.1560851960630076e-6,  -4.5516802628155526e-6,
       1.4236576649271475e-6,   1.8803149082089664e-14, -2.6623403898929211e-7,
       1.5950642189595716e-7,   -4.7187514673841102e-8, -6.5107872958755177e-17,
       7.9795091026746235e-9},
      {3.0249124160905891e-2,  2.4817436002649977e-3,  -4.9939134373457022e-2,
       5.9915643009307869e-2,  -3.2483207601623391e-2, -5.7212968652103441e-6,
       1.5085251778569354e-2,  -1.3261324005088445e-2, 5.5515262632426148e-3,
       3.0263182257030016e-8,  -1.7229548406756723e-3, 1.2893570099929637e-3,
       -4.6845138348319876e-4, -1.830259937893045e-10, 1.1449739014822654e-4,
       -7.7378565221244477e-5, 2.5625836246985201e-5,  1.0766165333192814e-12,
       -5.3246809282422621e-6, 3.349634863064464e-6,   -1.0381253128684018e-6,
       -5.608909920621128e-15, 1.9150821930676591e-7,  -1.1418365800203486e-7,
       3.3654425209171788e-8},
      {-9.9051020880159045e-2, 1.7954011706123486e-1,   -1.2989606383463778e-1,
       -3.1478872752284357e-5, 9.0510635276848131e-2,   -9.2828824411184397e-2,
       4.4412112839877808e-2,  2.7779236316835888e-7,   -1.7229543805449697e-2,
       1.4182925050891573e-2,  -5.6214161633747336e-3,  -2.39598509186381e-9,
       1.6029634366079908e-3,  -1.1606784674435773e-3,  4.1001337768153873e-4,
       1.8365800754090661e-11, -9.5844256563655903e-5,  6.3643062337764708e-5,
       -2.076250624489065e-5,  -1.1806020912804483e-13, 4.2131808239120649e-6,
       -2.6262241337012467e-6, 8.0770620494930662e-7,   6.0125912123632725e-16,
       -1.4729737374018841e-7},
      {-1.9994542198219728e-1, -1.5056113040026424e-2, 3.6470239469348489e-1,
       -4.6435192311733545e-1, 2.6640934719197893e-1,  3.4038266027147191e-5,
       -1.3784338709329624e-1, 1.276467178337056e-1,   -5.6213828755200985e-2,
       -1.753150885483011e-7,  1.9235592956768113e-2,  -1.5088821281095315e-2,
       5.7401854451350123e-3,  1.0622382710310225e-9,  -1.5335082692563998e-3,
       1.0819320643228214e-3,  -3.7372510193945659e-4, -6.6170909729031985e-12,
       8.4263617380909628e-5,  -5.5150706827483479e-5, 1.7769536448348069e-5,
       3.8827923210205533e-14, -3.53513697488768e-6,   2.1865832130045269e-6,
       -6.6812849447625594e-7},
      {7.2438608504029431e-1,   -1.3918010932653375,    1.0654143352413968,
       1.876173868950258e-4,    -8.2705501176152696e-1, 8.9352433347828414e-1,
       -4.4971003995291339e-1,  -1.6107401567546652e-6, 1.9235590165271091e-1,
       -1.6597702160042609e-1,  6.8882222681814333e-2,  1.3910091724608687e-8,
       -2.146911561508663e-2,   1.6228980898865892e-2,  -5.9796016172584256e-3,
       -1.1287469112826745e-10, 1.5167451119784857e-3,  -1.0478634293553899e-3,
       3.5539072889126421e-4,   8.1704322111801517e-13, -7.7773013442452395e-5,
       5.0291413897007722e-5,   -1.6035083867000518e-5, 1.2469354315487605e-14,
       3.1369106244517615e-6},
      {1.6668949727276811,     1.165462765994632e-1,   -3.3288393225018906,
       4.4692325482864037,     -2.6977693045875807,    -2.600667859891061e-4,
       1.5389017615694539,     -1.4937962361134612,    6.8881964633233148e-1,
       1.3077482004552385e-6,  -2.5762963325596288e-1, 2.1097676102125449e-1,
       -8.3714408359219882e-2, -7.7920428881354753e-9, 2.4267923064833599e-2,
       -1.7813678334552311e-2, 6.3970330388900056e-3,  4.9430807090480523e-11,
       -1.5554602758465635e-3, 1.0561196919903214e-3,  -3.5277184460472902e-4,
       9.3002334645022459e-14, 7.5285855026557172e-5,  -4.8186515569156351e-5,
       1.5227271505597605e-5},
      {-6.6188298861372935,    1.3397985455142589e+1,  -1.0789350606845146e+1,
       -1.4352254537875018e-3, 9.2333694596189809,     -1.0456552819547769e+1,
       5.5105526029033471,     1.2024439690716742e-5,  -2.5762961164755816,
       2.3207442745387179,     -1.0045728797216284,    -1.0207833290021914e-7,
       3.3975092171169466e-1,  -2.6720517450757468e-1, 1.0235252851562706e-1,
       8.4329730484871625e-10, -2.7998284958442595e-2, 2.0066274144976813e-2,
       -7.0554368915086242e-3, 1.9402238183698188e-12, 1.6562888105449611e-3,
       -1.1082898580743683e-3, 3.654545161310169e-4,   -5.1290032026971794e-11,
       -7.6340103696869031e-5},
      {-1.7112706061976095e+1, -1.1208044642899116,    3.7131966511885444e+1,
       -5.2298271025348962e+1, 3.3058589696624618e+1,  2.4791298976200222e-3,
       -2.061089403411526e+1,  2.088672775145582e+1,   -1.0045703956517752e+1,
       -1.2238783449063012e-5, 4.0770134274221141,     -3.473667358470195,
       1.4329352617312006,     7.1359914411879712e-8,  -4.4797257159115612e-1,
       3.4112666080644461e-1,  -1.2699786326594923e-1, -2.8953677269081528e-10,
       3.3125776278259863e-2,  -2.3274087021036101e-2, 8.0399993503648882e-3,
       -1.177805216235265e-9,  -1.8321624891071668e-3, 1.2108282933588665e-3,
       -3.9479941246822517e-4},
      {7.389033153567425e+1,   -1.5680141270402273e+2, 1.322177542759164e+2,
       1.3692876877324546e-2,  -1.2366496885920151e+2, 1.4620689391062729e+2,
       -8.0365587724865346e+1, -1.1259851148881298e-4, 4.0770132196179938e+1,
       -3.8210340013273034e+1, 1.719522294277362e+1,   9.3519707955168356e-7,
       -6.2716159907747034,    5.1168999071852637,     -2.0319658112299095,
       -4.9507215582761543e-9, 5.9626397294332597e-1,  -4.4220765337238094e-1,
       1.6079998700166273e-1,  -2.4733786203223402e-8, -4.0307574759979762e-2,
       2.7849050747097869e-2,  -9.4751858992054221e-3, 6.419922235909132e-6,
       2.1250180774699461e-3},
      {2.1216837098382522e+2,  1.3107863022633868e+1,  -4.9698285932871748e+2,
       7.3121595266969204e+2,  -4.8213821720890847e+2, -2.8817248692894889e-2,
       3.2616720302947102e+2,  -3.4389340280087117e+2, 1.7195193870816232e+2,
       1.4038077378096158e-4,  -7.52594195897599e+1,   6.651969984520934e+1,
       -2.8447519748152462e+1, -7.613702615875391e-7,  9.5402237105304373,
       -7.5175301113311376,    2.8943997568871961,     -4.6612194999538201e-7,
       -8.0615149598794088e-1, 5.8483006570631029e-1,  -2.0845408972964956e-1,
       1.4765818959305817e-4,  5.1000433863753019e-2,  -3.3066252141883665e-2,
       1.5109265210467774e-2},
      {-9.8959643098322368e+2, 2.1925555360905233e+3,  -1.9283586782723356e+3,
       -1.5925738122215253e-1, 1.9569985945919857e+3,  -2.4072514765081556e+3,
       1.3756149959336496e+3,  1.2920735237496668e-3,  -7.525941715948055e+2,
       7.3171668742208716e+2,  -3.4137023466220065e+2, -9.9857390260608043e-6,
       1.3356313181291573e+2,  -1.1276295161252794e+2, 4.6310396098204458e+1,
       -7.9237387133614756e-6, -1.4510726927018646e+1, 1.1111771248100563e+1,
       -4.1690817945270892,    3.1008219800117808e-3,  1.1220095449981468,
       -7.6052379926149916e-1, 3.6262236505085254e-1,  2.216867741940747e-1,
       4.8683443692930507e-1}};
  int k, n, sgn;
  int maxpow = 0;
  static const accscalar_t MACHEP = std::is_same<accscalar_t, double>::value
      ? 1.11022302462515654042E-16
      : 5.9604644775390625E-8;
  accscalar_t lambda = x / a;
  accscalar_t sigma = (x - a) / a;
  accscalar_t eta, res, ck, ckterm, term, absterm;
  accscalar_t absoldterm = INFINITY;
  accscalar_t etapow[25] = {1};
  accscalar_t sum = 0;
  accscalar_t afac = 1;

  if (igam) {
    sgn = -1;
  } else {
    sgn = 1;
  }
  if (lambda > 1) {
    eta = Numerics<accscalar_t>::sqrt(
        -2 * (Numerics<accscalar_t>::log1p(sigma) - sigma));
  } else if (lambda < 1) {
    eta = -Numerics<accscalar_t>::sqrt(
        -2 * (Numerics<accscalar_t>::log1p(sigma) - sigma));
  } else {
    eta = 0;
  }
  res = 0.5 *
      Numerics<accscalar_t>::erfc(
            sgn * eta * Numerics<accscalar_t>::sqrt(a / 2));

  for (k = 0; k < 25; k++) {
    ck = d[k][0];
    for (n = 1; n < 25; n++) {
      if (n > maxpow) {
        etapow[n] = eta * etapow[n - 1];
        maxpow += 1;
      }
      ckterm = d[k][n] * etapow[n];
      ck += ckterm;
      if (std::fabs(ckterm) < MACHEP * std::fabs(ck)) {
        break;
      }
    }
    term = ck * afac;
    absterm = std::fabs(term);
    if (absterm > absoldterm) {
      break;
    }
    term = ck * afac;
    absterm = std::fabs(term);
    if (absterm > absoldterm) {
      break;
    }
    sum += term;
    if (absterm < MACHEP * std::fabs(sum)) {
      break;
    }
    absoldterm = absterm;
    afac /= a;
  }
  res += sgn * Numerics<scalar_t>::exp(-0.5 * a * eta * eta) * sum /
      Numerics<scalar_t>::sqrt(2 * 3.1415926535 * a);

  return res;
}
template <typename scalar_t>
static inline scalar_t _igamc_helper_continued_fraction(
    scalar_t a,
    scalar_t x) {
  // Compute igamc using DLMF 8.9.2. [igam1]

  using accscalar_t = acc_type<scalar_t>;
  int i;
  accscalar_t ans, ax, c, yc, r, t, y, z;
  accscalar_t pk, pkm1, pkm2, qk, qkm1, qkm2;
  static const int MAXITER = 2000;
  static const accscalar_t MACHEP = std::is_same<accscalar_t, double>::value
      ? 1.11022302462515654042E-16
      : 5.9604644775390625E-8;
  static const accscalar_t BIG = std::is_same<accscalar_t, double>::value
      ? 4.503599627370496e15
      : 16777216.;
  static const accscalar_t BIGINV = std::is_same<accscalar_t, double>::value
      ? 2.22044604925031308085e-16
      : 5.9604644775390625E-8;

  ax = _igam_helper_fac(a, x);
  if (ax == 0.0) {
    return 0.0;
  }

  /* continued fraction */
  y = 1.0 - a;
  z = x + y + 1.0;
  c = 0.0;
  pkm2 = 1.0;
  qkm2 = x;
  pkm1 = x + 1.0;
  qkm1 = z * x;
  ans = pkm1 / qkm1;

  for (i = 0; i < MAXITER; i++) {
    c += 1.0;
    y += 1.0;
    z += 2.0;
    yc = y * c;
    pk = pkm1 * z - pkm2 * yc;
    qk = qkm1 * z - qkm2 * yc;
    if (qk != 0) {
      r = pk / qk;
      t = std::fabs((ans - r) / r);
      ans = r;
    } else {
      t = 1.0;
    }
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;
    if (std::fabs(pk) > BIG) {
      pkm2 *= BIGINV;
      pkm1 *= BIGINV;
      qkm2 *= BIGINV;
      qkm1 *= BIGINV;
    }
    if (t <= MACHEP) {
      break;
    }
  }
  return ans * ax;
}

template <typename scalar_t>
static inline scalar_t calc_igammac(scalar_t a, scalar_t x) {
  /* the calculation of the regularized upper incomplete gamma function
   * is done differently based on the values of a and x:
   * - if x and/or a is at the boundary of defined region, then assign the
   *   result at the boundary
   * - if a is large and a ~ x, then using Uniform Asymptotic Expansions for
   *   Large Parameter (see DLMF 8.12.4 [igam1])
   * - if x > 1.1 and x < a, using the substraction from the regularized lower
   *   incomplete gamma
   * - otherwise, calculate the series from [igam2] eq (5)
   */

  using accscalar_t = acc_type<scalar_t>;
  accscalar_t absxma_a;

  static const accscalar_t SMALL = 20.0;
  static const accscalar_t LARGE = 200.0;
  static const accscalar_t SMALLRATIO = 0.3;
  static const accscalar_t LARGERATIO = 4.5;

  if ((x < 0) || (a < 0)) {
    // out of defined-region of the function
    return std::numeric_limits<accscalar_t>::quiet_NaN();
  } else if (a == 0) {
    if (x > 0) {
      return 0.0;
    } else {
      return std::numeric_limits<accscalar_t>::quiet_NaN();
    }
  } else if (x == 0) {
    return 1.0;
  } else if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(a))) {
    if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(x))) {
      return std::numeric_limits<accscalar_t>::quiet_NaN();
    }
    return 1.0;
  } else if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(x))) {
    return 0.0;
  }

  absxma_a = std::fabs(static_cast<accscalar_t>(x - a)) / a;
  if ((a > SMALL) && (a < LARGE) && (absxma_a < SMALLRATIO)) {
    return _igam_helper_asymptotic_series(a, x, 0);
  } else if ((a > LARGE) && (absxma_a < LARGERATIO / ::sqrt(a))) {
    return _igam_helper_asymptotic_series(a, x, 0);
  }

  if (x > 1.1f) {
    if (x < a) {
      return 1.0 - _igam_helper_series(a, x);
    } else {
      return _igamc_helper_continued_fraction(a, x);
    }
  } else if (x <= 0.5) {
    if (-0.4 / Numerics<scalar_t>::log(x) < a) {
      return 1.0 - _igam_helper_series(a, x);
    } else {
      return _igamc_helper_series(a, x);
    }
  } else {
    if (x * 1.1 < a) {
      return 1.0 - _igam_helper_series(a, x);
    } else {
      return _igamc_helper_series(a, x);
    }
  }
}

template <typename scalar_t>
static inline scalar_t calc_igamma(scalar_t a, scalar_t x) {
  /* the calculation of the regularized lower incomplete gamma function
   * is done differently based on the values of a and x:
   * - if x and/or a is at the boundary of defined region, then assign the
   *   result at the boundary
   * - if a is large and a ~ x, then using Uniform Asymptotic Expansions for
   *   Large Parameter (see DLMF 8.12.3 [igam1])
   * - if x > 1 and x > a, using the substraction from the regularized upper
   *   incomplete gamma
   * - otherwise, calculate the series from [igam2] eq (4)
   */

  using accscalar_t = acc_type<scalar_t>;
  accscalar_t absxma_a;
  static const accscalar_t SMALL = 20.0;
  static const accscalar_t LARGE = 200.0;
  static const accscalar_t SMALLRATIO = 0.3;
  static const accscalar_t LARGERATIO = 4.5;

  // boundary values following SciPy
  if ((x < 0) || (a < 0)) {
    // out of defined-region of the function
    return std::numeric_limits<accscalar_t>::quiet_NaN();
  } else if (a == 0) {
    if (x > 0) {
      return 1.0;
    } else {
      return std::numeric_limits<accscalar_t>::quiet_NaN();
    }
  } else if (x == 0) {
    return 0.0; // zero integration limit
  } else if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(a))) {
    if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(x))) {
      return std::numeric_limits<accscalar_t>::quiet_NaN();
    }
    return 0.0;
  } else if (Numerics<accscalar_t>::isinf(static_cast<accscalar_t>(x))) {
    return 1.0;
  }

  /* Asymptotic regime where a ~ x. */
  absxma_a = std::fabs(static_cast<accscalar_t>(x - a)) / a;
  if ((a > SMALL) && (a < LARGE) && (absxma_a < SMALLRATIO)) {
    return _igam_helper_asymptotic_series(a, x, 1);
  } else if ((a > LARGE) && (absxma_a < LARGERATIO / ::sqrt(a))) {
    return _igam_helper_asymptotic_series(a, x, 1);
  }

  if ((x > 1.0) && (x > a)) {
    return 1.0 - calc_igammac(a, x);
  }

  return _igam_helper_series(a, x);
}

template <typename scalar_t>
static inline scalar_t calc_digamma(scalar_t in) {
  // [C++ Standard Reference: Gamma Function]
  // https://en.cppreference.com/w/cpp/numeric/math/tgamma
  using accscalar_t = acc_type<scalar_t>;
  static const double PI_f64 = 3.14159265358979323846;
  const accscalar_t PSI_10 = 2.25175258906672110764;
  const accscalar_t A[] = {
      8.33333333333333333333E-2,
      -2.10927960927960927961E-2,
      7.57575757575757575758E-3,
      -4.16666666666666666667E-3,
      3.96825396825396825397E-3,
      -8.33333333333333333333E-3,
      8.33333333333333333333E-2,
  };

  accscalar_t x = static_cast<accscalar_t>(in);
  if (x == 0) {
    // As per C++ standard for gamma related functions and SciPy,
    // If the argument is ±0, ±∞ is returned
    return std::copysign(static_cast<scalar_t>(INFINITY), -x);
  }

  bool x_is_integer = x == ::trunc(x);
  accscalar_t result = 0;
  if (x < 0) {
    if (x_is_integer) {
      // As per C++ standard for gamma related functions and SciPy,
      // If the argument is a negative integer, NaN is returned
      return static_cast<scalar_t>(NAN);
    }
    // Extracts the fractional part of x as r, since tan(pi * r) is more
    // numerically accurate than tan(pi * x). While these operations are
    // mathematically equivalent since both x and r are in radians and tan() has
    // a periodicity of pi, in practice the computation of pi * x is a source of
    // error (when |x| > 1).
    double q, r;
    r = std::modf(static_cast<double>(x), &q);
    result = static_cast<accscalar_t>(-PI_f64 / std::tan(PI_f64 * r));
    x = 1 - x;
  }

  while (x < 10) {
    result -= 1 / x;
    x += 1;
  }
  if (x == 10) {
    return static_cast<scalar_t>(result + PSI_10);
  }

  accscalar_t y = 0;
  if (x < 1.0e17f) {
    accscalar_t z = 1 / (x * x);

    accscalar_t polevl_result = 0;
    for (int i = 0; i <= 6; i++) {
      polevl_result = polevl_result * z + A[i];
    }
    y = z * polevl_result;
  }

  return static_cast<scalar_t>(
      std::log(x) - (static_cast<accscalar_t>(0.5) / x) - y + result);
}

template <typename scalar_t>
static inline scalar_t calc_trigamma(scalar_t in) {
  using accscalar_t = acc_type<scalar_t>;
  const accscalar_t PI = 3.14159265358979323846;
  accscalar_t x = static_cast<accscalar_t>(in);
  accscalar_t sign = +1;
  accscalar_t result = 0;
  if (x < 0.5f) {
    sign = -1;
    accscalar_t sin_pi_x = std::sin(PI * x);
    result -= (PI * PI) / (sin_pi_x * sin_pi_x);
    x = 1 - x;
  }
  for (int i = 0; i < 6; ++i) {
    result += 1 / (x * x);
    x += 1;
  }
  const accscalar_t one = static_cast<scalar_t>(1);
  const accscalar_t ixx = 1 / (x * x);
  result += (1 + 1 / (2 * x) +
             ixx * (one / 6 - ixx * (one / 30 - ixx * (one / 42)))) /
      x;
  return static_cast<scalar_t>(sign * result);
}

template <typename scalar_t>
static inline scalar_t calc_gcd(scalar_t a_in, scalar_t b_in) {
  scalar_t a = Numerics<scalar_t>::abs(a_in);
  scalar_t b = Numerics<scalar_t>::abs(b_in);
  while (a != 0) {
    scalar_t c = a;
    a = b % a;
    b = c;
  }
  return b;
}

/*
 * For licensing information and documentation, please refer to the the cpu
 * implementation located in "ATen/native/Math.h".
 */
template <typename scalar_t>
static inline scalar_t chbevl(scalar_t _x, const scalar_t array[], size_t len) {
  static_assert(
      !std::is_same<scalar_t, Half>() && !std::is_same<scalar_t, BFloat16>(),
      "don't instantiate with low precision type");
  using accscalar_t = acc_type<scalar_t>;

  accscalar_t x = static_cast<accscalar_t>(_x);
  accscalar_t b0, b1, b2;

  b0 = static_cast<accscalar_t>(array[0]);
  b1 = 0;

  for (size_t i = 1; i < len; ++i) {
    b2 = b1;
    b1 = b0;
    b0 = x * b1 - b2 + static_cast<accscalar_t>(array[i]);
  }

  return static_cast<scalar_t>(0.5 * (b0 - b2));
}

/*
 * For licensing information and documentation, please refer to the the cpu
 * implementation located in "ATen/native/Math.h".
 */
template <typename T>
inline std::tuple<const T*, size_t> chebyshev_coefficients_i0e_A() {
  /* Chebyshev coefficients for exp(-x) I0(x)
   * in the interval [0,8].
   *
   * lim(x->0){ exp(-x) I0(x) } = 1.
   */
  static const T coefficients[] = {
      -4.41534164647933937950E-18, 3.33079451882223809783E-17,
      -2.43127984654795469359E-16, 1.71539128555513303061E-15,
      -1.16853328779934516808E-14, 7.67618549860493561688E-14,
      -4.85644678311192946090E-13, 2.95505266312963983461E-12,
      -1.72682629144155570723E-11, 9.67580903537323691224E-11,
      -5.18979560163526290666E-10, 2.65982372468238665035E-9,
      -1.30002500998624804212E-8,  6.04699502254191894932E-8,
      -2.67079385394061173391E-7,  1.11738753912010371815E-6,
      -4.41673835845875056359E-6,  1.64484480707288970893E-5,
      -5.75419501008210370398E-5,  1.88502885095841655729E-4,
      -5.76375574538582365885E-4,  1.63947561694133579842E-3,
      -4.32430999505057594430E-3,  1.05464603945949983183E-2,
      -2.37374148058994688156E-2,  4.93052842396707084878E-2,
      -9.49010970480476444210E-2,  1.71620901522208775349E-1,
      -3.04682672343198398683E-1,  6.76795274409476084995E-1};

  return std::make_tuple(coefficients, 30);
}

template <typename T>
inline std::tuple<const T*, size_t> chebyshev_coefficients_i0e_B() {
  /* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
   * in the inverted interval [8,infinity].
   *
   * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
   */
  static const T coefficients[] = {
      -7.23318048787475395456E-18, -4.83050448594418207126E-18,
      4.46562142029675999901E-17,  3.46122286769746109310E-17,
      -2.82762398051658348494E-16, -3.42548561967721913462E-16,
      1.77256013305652638360E-15,  3.81168066935262242075E-15,
      -9.55484669882830764870E-15, -4.15056934728722208663E-14,
      1.54008621752140982691E-14,  3.85277838274214270114E-13,
      7.18012445138366623367E-13,  -1.79417853150680611778E-12,
      -1.32158118404477131188E-11, -3.14991652796324136454E-11,
      1.18891471078464383424E-11,  4.94060238822496958910E-10,
      3.39623202570838634515E-9,   2.26666899049817806459E-8,
      2.04891858946906374183E-7,   2.89137052083475648297E-6,
      6.88975834691682398426E-5,   3.36911647825569408990E-3,
      8.04490411014108831608E-1};

  return std::make_tuple(coefficients, 25);
}

template <typename scalar_t>
static inline scalar_t calc_i0(scalar_t _x) {
  static_assert(
      !std::is_same<scalar_t, Half>() && !std::is_same<scalar_t, BFloat16>(),
      "don't instantiate with low precision type");
  // Upcast input for numerical accuracy purposes
  // Needed for accurate results if input is bfloat16 or float16
  scalar_t x = Numerics<scalar_t>::abs(_x);

  if (x <= scalar_t{8.0}) {
    auto coeff_pair = chebyshev_coefficients_i0e_A<scalar_t>();
    auto A = std::get<0>(coeff_pair);
    auto len = std::get<1>(coeff_pair);
    scalar_t y = (x / scalar_t{2.0}) - scalar_t{2.0};
    return (std::exp(x) * chbevl(y, A, len));
  }

  auto coeff_pair = chebyshev_coefficients_i0e_B<scalar_t>();
  auto B = std::get<0>(coeff_pair);
  auto len = std::get<1>(coeff_pair);
  return (
      std::exp(x) * chbevl(scalar_t{32.0} / x - scalar_t{2.0}, B, len) /
      std::sqrt(x));
}

/*
 * This function is derived from the implementation of the i0e function in the
 * Cephes Math Library. See note [3-Clause BSD License for the Cephes Math
 * Library].
 *
 * Computes an approximation of the exponentially scaled zeroth order modified
 * Bessel function of the first kind. The approximation is actually two
 * (sub)approximations, both using a Chebyshev polynomial expansion. One
 * approximates the function over [0, 8], and the other over (8, infinity). This
 * function takes the absolute value of all inputs to convert them into the
 * domain of the approximation.
 */
template <typename scalar_t>
static inline scalar_t calc_i0e(scalar_t _x) {
  scalar_t x = Numerics<scalar_t>::abs(_x);

  if (x <= scalar_t{8.0}) {
    auto coeff_pair = chebyshev_coefficients_i0e_A<scalar_t>();
#ifdef SYCL_DEVICE_ONLY
    auto A = std::get<0>(coeff_pair);
    auto len = std::get<1>(coeff_pair);
#else
    auto A = std::get<0>(coeff_pair);
    auto len = std::get<1>(coeff_pair);
#endif
    scalar_t y = (x / scalar_t{2.0}) - scalar_t{2.0};
    return chbevl(y, A, len);
  }

  auto coeff_pair = chebyshev_coefficients_i0e_B<scalar_t>();
#ifdef SYCL_DEVICE_ONLY
  auto B = std::get<0>(coeff_pair);
  auto len = std::get<1>(coeff_pair);
#else
  auto B = std::get<0>(coeff_pair);
  auto len = std::get<1>(coeff_pair);
#endif
  return chbevl(scalar_t{32.0} / x - scalar_t{2.0}, B, len) / std::sqrt(x);
}

// Upcast bfloat16 input to float for numerical accuracy purposes
static inline c10::BFloat16 calc_i0e(c10::BFloat16 a) {
  return calc_i0e(static_cast<float>(a));
}

template <typename T>
inline typename std::enable_if<
    std::is_same<double, T>::value,
    std::tuple<const T*, size_t>>::type
chebyshev_coefficients_i1e_A() {
  /* Chebyshev coefficients for exp(-x) I1(x)
   * in the interval [0,8].
   *
   * lim(x->0){ exp(-x) I1(x) / x } = 1/2.
   */
  static const T coefficients[] = {
      2.77791411276104639959E-18, -2.11142121435816608115E-17,
      1.55363195773620046921E-16, -1.10559694773538630805E-15,
      7.60068429473540693410E-15, -5.04218550472791168711E-14,
      3.22379336594557470981E-13, -1.98397439776494371520E-12,
      1.17361862988909016308E-11, -6.66348972350202774223E-11,
      3.62559028155211703701E-10, -1.88724975172282928790E-9,
      9.38153738649577178388E-9,  -4.44505912879632808065E-8,
      2.00329475355213526229E-7,  -8.56872026469545474066E-7,
      3.47025130813767847674E-6,  -1.32731636560394358279E-5,
      4.78156510755005422638E-5,  -1.61760815825896745588E-4,
      5.12285956168575772895E-4,  -1.51357245063125314899E-3,
      4.15642294431288815669E-3,  -1.05640848946261981558E-2,
      2.47264490306265168283E-2,  -5.29459812080949914269E-2,
      1.02643658689847095384E-1,  -1.76416518357834055153E-1,
      2.52587186443633654823E-1};

  return std::make_tuple(coefficients, 29);
}

template <typename T>
inline typename std::
    enable_if<std::is_same<float, T>::value, std::tuple<const T*, size_t>>::type
    chebyshev_coefficients_i1e_A() {
  /* Chebyshev coefficients for exp(-x) I1(x)
   * in the interval [0,8].
   *
   * lim(x->0){ exp(-x) I1(x) / x } = 1/2.
   */
  static const T coeff[] = {
      9.38153738649577178388E-9f,
      -4.44505912879632808065E-8f,
      2.00329475355213526229E-7f,
      -8.56872026469545474066E-7f,
      3.47025130813767847674E-6f,
      -1.32731636560394358279E-5f,
      4.78156510755005422638E-5f,
      -1.61760815825896745588E-4f,
      5.12285956168575772895E-4f,
      -1.51357245063125314899E-3f,
      4.15642294431288815669E-3f,
      -1.05640848946261981558E-2f,
      2.47264490306265168283E-2f,
      -5.29459812080949914269E-2f,
      1.02643658689847095384E-1f,
      -1.76416518357834055153E-1f,
      2.52587186443633654823E-1f};
  return std::make_tuple(coeff, 17);
};

template <typename T>
inline typename std::enable_if<
    std::is_same<double, T>::value,
    std::tuple<const T*, size_t>>::type
chebyshev_coefficients_i1e_B() {
  /* Chebyshev coefficients for exp(-x) sqrt(x) I1(x)
   * in the inverted interval [8,infinity].
   *
   * lim(x->inf){ exp(-x) sqrt(x) I1(x) } = 1/sqrt(2pi).
   */
  static const T coefficients[] = {
      7.51729631084210481353E-18,  4.41434832307170791151E-18,
      -4.65030536848935832153E-17, -3.20952592199342395980E-17,
      2.96262899764595013876E-16,  3.30820231092092828324E-16,
      -1.88035477551078244854E-15, -3.81440307243700780478E-15,
      1.04202769841288027642E-14,  4.27244001671195135429E-14,
      -2.10154184277266431302E-14, -4.08355111109219731823E-13,
      -7.19855177624590851209E-13, 2.03562854414708950722E-12,
      1.41258074366137813316E-11,  3.25260358301548823856E-11,
      -1.89749581235054123450E-11, -5.58974346219658380687E-10,
      -3.83538038596423702205E-9,  -2.63146884688951950684E-8,
      -2.51223623787020892529E-7,  -3.88256480887769039346E-6,
      -1.10588938762623716291E-4,  -9.76109749136146840777E-3,
      7.78576235018280120474E-1};

  return std::make_tuple(coefficients, 25);
}

template <typename T>
inline typename std::
    enable_if<std::is_same<float, T>::value, std::tuple<const T*, size_t>>::type
    chebyshev_coefficients_i1e_B() {
  /* Chebyshev coefficients for exp(-x) sqrt(x) I1(x)
   * in the inverted interval [8,infinity].
   *
   * lim(x->inf){ exp(-x) sqrt(x) I1(x) } = 1/sqrt(2pi).
   */
  static const T coeff[] = {
      -3.83538038596423702205E-9f,
      -2.63146884688951950684E-8f,
      -2.51223623787020892529E-7f,
      -3.88256480887769039346E-6f,
      -1.10588938762623716291E-4f,
      -9.76109749136146840777E-3f,
      7.78576235018280120474E-1f};

  return std::make_tuple(coeff, 7);
};

template <typename scalar_t>
static inline scalar_t calc_i1(scalar_t _x) {
  const auto x = Numerics<scalar_t>::abs(_x);
  if (x <= scalar_t{8.0}) {
    auto coeff_pair = chebyshev_coefficients_i1e_A<scalar_t>();
    auto A = std::get<0>(coeff_pair);
    auto len = std::get<1>(coeff_pair);
    scalar_t y = x / scalar_t{2.0} - scalar_t{2.0};
    const scalar_t out = std::exp(x) * x * chbevl(y, A, len);
    return (_x < scalar_t{0.0}) ? -out : out;
  }

  auto coeff_pair = chebyshev_coefficients_i1e_B<scalar_t>();
  auto B = std::get<0>(coeff_pair);
  auto len = std::get<1>(coeff_pair);
  const scalar_t out =
      (std::exp(x) * chbevl(scalar_t{32.0} / x - scalar_t{2.0}, B, len)) /
      std::sqrt(x);
  return (_x < scalar_t{0.0}) ? -out : out;
}

template <typename scalar_t>
static inline scalar_t calc_i1e(scalar_t _x) {
  const auto x = Numerics<scalar_t>::abs(_x);
  if (x <= scalar_t{8.0}) {
    auto coeff_pair = chebyshev_coefficients_i1e_A<scalar_t>();
    auto A = std::get<0>(coeff_pair);
    auto len = std::get<1>(coeff_pair);
    const scalar_t y = x / scalar_t{2.0} - scalar_t{2.0};
    const scalar_t out = chbevl(y, A, len) * x;
    return (_x < scalar_t{0.0}) ? -out : out;
  }

  auto coeff_pair = chebyshev_coefficients_i1e_B<scalar_t>();
  auto B = std::get<0>(coeff_pair);
  auto len = std::get<1>(coeff_pair);
  const scalar_t out =
      chbevl(scalar_t{32.0} / x - scalar_t{2.0}, B, len) / std::sqrt(x);
  return (_x < scalar_t{0.0}) ? -out : out;
}

template <typename scalar_t>
static inline scalar_t calc_polygamma(scalar_t x, int n) {
  // already blocked if n <= 1
  const auto one = scalar_t{1};
  return ((n % 2) ? one : -one) *
      std::exp(std::lgamma(static_cast<scalar_t>(n) + one)) *
      zeta<scalar_t>(static_cast<scalar_t>(n + 1), x);
}
/*
 * This function is derived from the implementation of the digamma function in
 * the Cephes Math Library. See note [3-Clause BSD License for the Cephes Math
 * Library].
 *
 * Evaluates polynomial of degree N:
 *
 *                     2          N
 * y  =  C  + C x + C x  +...+ C x
 *        0    1     2          N
 *
 * Coefficients are stored in reverse order:
 *
 * coef[0] = C  , ..., coef[N] = C  .
 *            N                   0
 */
template <typename T>
static inline T polevl(const T x, const T A[], size_t len) {
  T result = 0;
  for (size_t i = 0; i <= len; i++) {
    result = result * x + A[i];
  }
  return result;
}

/*
 * This function is derived from the implementation of the ndtri function in the
 * Cephes Math Library. See note [3-Clause BSD License for the Cephes Math
 * Library].
 *
 * Computes the argument, x, for which the area under the Gaussian probability
 * density function (integrated from minus infinity to x) is equal to y.
 */
template <typename scalar_t>
static inline scalar_t calc_ndtri(scalar_t y0) {
  /* sqrt(2pi) */
  constexpr scalar_t s2pi = 2.50662827463100050242E0;
  constexpr scalar_t one = 1;
  constexpr scalar_t zero = 0;

  /* approximation for 0 <= |y - 0.5| <= 3/8 */
  static const scalar_t P0[5] = {
      -5.99633501014107895267E1,
      9.80010754185999661536E1,
      -5.66762857469070293439E1,
      1.39312609387279679503E1,
      -1.23916583867381258016E0,
  };
  static const scalar_t Q0[9] = {
      1.00000000000000000000E0,
      1.95448858338141759834E0,
      4.67627912898881538453E0,
      8.63602421390890590575E1,
      -2.25462687854119370527E2,
      2.00260212380060660359E2,
      -8.20372256168333339912E1,
      1.59056225126211695515E1,
      -1.18331621121330003142E0,
  };

  /* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
   * i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
   */
  static const scalar_t P1[9] = {
      4.05544892305962419923E0,
      3.15251094599893866154E1,
      5.71628192246421288162E1,
      4.40805073893200834700E1,
      1.46849561928858024014E1,
      2.18663306850790267539E0,
      -1.40256079171354495875E-1,
      -3.50424626827848203418E-2,
      -8.57456785154685413611E-4,
  };
  static const scalar_t Q1[9] = {
      1.00000000000000000000E0,
      1.57799883256466749731E1,
      4.53907635128879210584E1,
      4.13172038254672030440E1,
      1.50425385692907503408E1,
      2.50464946208309415979E0,
      -1.42182922854787788574E-1,
      -3.80806407691578277194E-2,
      -9.33259480895457427372E-4,
  };

  /* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
   * i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
   */

  static const scalar_t P2[9] = {
      3.23774891776946035970E0,
      6.91522889068984211695E0,
      3.93881025292474443415E0,
      1.33303460815807542389E0,
      2.01485389549179081538E-1,
      1.23716634817820021358E-2,
      3.01581553508235416007E-4,
      2.65806974686737550832E-6,
      6.23974539184983293730E-9,
  };
  static const scalar_t Q2[9] = {
      1.00000000000000000000E0,
      6.02427039364742014255E0,
      3.67983563856160859403E0,
      1.37702099489081330271E0,
      2.16236993594496635890E-1,
      1.34204006088543189037E-2,
      3.28014464682127739104E-4,
      2.89247864745380683936E-6,
      6.79019408009981274425E-9,
  };

  if (y0 == zero) {
    return Numerics<scalar_t>::lower_bound();
  }
  if (y0 == one) {
    return Numerics<scalar_t>::upper_bound();
  }
  if (y0 < zero || y0 > one) {
    return std::numeric_limits<scalar_t>::quiet_NaN();
  }
  bool code = true;
  scalar_t y = y0;
  if (y > one - scalar_t{0.13533528323661269189}) { /* 0.135... = exp(-2) */
    y = one - y;
    code = false;
  }
  if (y > scalar_t{0.13533528323661269189}) {
    y = y - scalar_t{0.5};
    const scalar_t y2 = y * y;
    scalar_t x = y + y * (y2 * polevl(y2, P0, 4) / polevl(y2, Q0, 8));
    return (x * s2pi);
  }

  scalar_t x = ::sqrt(scalar_t{-2.0} * ::log(y));
  const scalar_t x0 = x - ::log(x) / x;

  const scalar_t z = one / x;
  scalar_t x1;
  if (x < scalar_t{8.0}) /* y > exp(-32) = 1.2664165549e-14 */
  {
    x1 = z * polevl(z, P1, 8) / polevl(z, Q1, 8);
  } else {
    x1 = z * polevl(z, P2, 8) / polevl(z, Q2, 8);
  }
  x = x0 - x1;
  if (code) {
    x = -x;
  }
  return x;
}
// Upcast bfloat16 input to float for numerical accuracy purposes
static inline c10::BFloat16 calc_ndtri(c10::BFloat16 a) {
  return calc_ndtri(static_cast<float>(a));
}

template <typename T>
static inline T erfcx_y100(T y100) {
  switch (static_cast<int>(y100)) {
    case 0: {
      T t = 2 * y100 - 1;
      return 0.70878032454106438663e-3 +
          (0.71234091047026302958e-3 +
           (0.35779077297597742384e-5 +
            (0.17403143962587937815e-7 +
             (0.81710660047307788845e-10 +
              (0.36885022360434957634e-12 + 0.15917038551111111111e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 1: {
      T t = 2 * y100 - 3;
      return 0.21479143208285144230e-2 +
          (0.72686402367379996033e-3 +
           (0.36843175430938995552e-5 +
            (0.18071841272149201685e-7 +
             (0.85496449296040325555e-10 +
              (0.38852037518534291510e-12 + 0.16868473576888888889e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 2: {
      T t = 2 * y100 - 5;
      return 0.36165255935630175090e-2 +
          (0.74182092323555510862e-3 +
           (0.37948319957528242260e-5 +
            (0.18771627021793087350e-7 +
             (0.89484715122415089123e-10 +
              (0.40935858517772440862e-12 + 0.17872061464888888889e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 3: {
      T t = 2 * y100 - 7;
      return 0.51154983860031979264e-2 +
          (0.75722840734791660540e-3 +
           (0.39096425726735703941e-5 +
            (0.19504168704300468210e-7 +
             (0.93687503063178993915e-10 +
              (0.43143925959079664747e-12 + 0.18939926435555555556e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 4: {
      T t = 2 * y100 - 9;
      return 0.66457513172673049824e-2 +
          (0.77310406054447454920e-3 +
           (0.40289510589399439385e-5 +
            (0.20271233238288381092e-7 +
             (0.98117631321709100264e-10 +
              (0.45484207406017752971e-12 + 0.20076352213333333333e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 5: {
      T t = 2 * y100 - 11;
      return 0.82082389970241207883e-2 +
          (0.78946629611881710721e-3 +
           (0.41529701552622656574e-5 +
            (0.21074693344544655714e-7 +
             (0.10278874108587317989e-9 +
              (0.47965201390613339638e-12 + 0.21285907413333333333e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 6: {
      T t = 2 * y100 - 13;
      return 0.98039537275352193165e-2 +
          (0.80633440108342840956e-3 +
           (0.42819241329736982942e-5 +
            (0.21916534346907168612e-7 +
             (0.10771535136565470914e-9 +
              (0.50595972623692822410e-12 + 0.22573462684444444444e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 7: {
      T t = 2 * y100 - 15;
      return 0.11433927298290302370e-1 +
          (0.82372858383196561209e-3 +
           (0.44160495311765438816e-5 +
            (0.22798861426211986056e-7 +
             (0.11291291745879239736e-9 +
              (0.53386189365816880454e-12 + 0.23944209546666666667e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 8: {
      T t = 2 * y100 - 17;
      return 0.13099232878814653979e-1 +
          (0.84167002467906968214e-3 +
           (0.45555958988457506002e-5 +
            (0.23723907357214175198e-7 +
             (0.11839789326602695603e-9 +
              (0.56346163067550237877e-12 + 0.25403679644444444444e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 9: {
      T t = 2 * y100 - 19;
      return 0.14800987015587535621e-1 +
          (0.86018092946345943214e-3 +
           (0.47008265848816866105e-5 +
            (0.24694040760197315333e-7 +
             (0.12418779768752299093e-9 +
              (0.59486890370320261949e-12 + 0.26957764568888888889e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 10: {
      T t = 2 * y100 - 21;
      return 0.16540351739394069380e-1 +
          (0.87928458641241463952e-3 +
           (0.48520195793001753903e-5 +
            (0.25711774900881709176e-7 +
             (0.13030128534230822419e-9 +
              (0.62820097586874779402e-12 + 0.28612737351111111111e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 11: {
      T t = 2 * y100 - 23;
      return 0.18318536789842392647e-1 +
          (0.89900542647891721692e-3 +
           (0.50094684089553365810e-5 +
            (0.26779777074218070482e-7 +
             (0.13675822186304615566e-9 +
              (0.66358287745352705725e-12 + 0.30375273884444444444e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 12: {
      T t = 2 * y100 - 25;
      return 0.20136801964214276775e-1 +
          (0.91936908737673676012e-3 +
           (0.51734830914104276820e-5 +
            (0.27900878609710432673e-7 +
             (0.14357976402809042257e-9 +
              (0.70114790311043728387e-12 + 0.32252476000000000000e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 13: {
      T t = 2 * y100 - 27;
      return 0.21996459598282740954e-1 +
          (0.94040248155366777784e-3 +
           (0.53443911508041164739e-5 +
            (0.29078085538049374673e-7 +
             (0.15078844500329731137e-9 +
              (0.74103813647499204269e-12 + 0.34251892320000000000e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 14: {
      T t = 2 * y100 - 29;
      return 0.23898877187226319502e-1 +
          (0.96213386835900177540e-3 +
           (0.55225386998049012752e-5 +
            (0.30314589961047687059e-7 +
             (0.15840826497296335264e-9 +
              (0.78340500472414454395e-12 + 0.36381553564444444445e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 15: {
      T t = 2 * y100 - 31;
      return 0.25845480155298518485e-1 +
          (0.98459293067820123389e-3 +
           (0.57082915920051843672e-5 +
            (0.31613782169164830118e-7 +
             (0.16646478745529630813e-9 +
              (0.82840985928785407942e-12 + 0.38649975768888888890e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 16: {
      T t = 2 * y100 - 33;
      return 0.27837754783474696598e-1 +
          (0.10078108563256892757e-2 +
           (0.59020366493792212221e-5 +
            (0.32979263553246520417e-7 +
             (0.17498524159268458073e-9 +
              (0.87622459124842525110e-12 + 0.41066206488888888890e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 17: {
      T t = 2 * y100 - 35;
      return 0.29877251304899307550e-1 +
          (0.10318204245057349310e-2 +
           (0.61041829697162055093e-5 +
            (0.34414860359542720579e-7 +
             (0.18399863072934089607e-9 +
              (0.92703227366365046533e-12 + 0.43639844053333333334e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 18: {
      T t = 2 * y100 - 37;
      return 0.31965587178596443475e-1 +
          (0.10566560976716574401e-2 +
           (0.63151633192414586770e-5 +
            (0.35924638339521924242e-7 +
             (0.19353584758781174038e-9 +
              (0.98102783859889264382e-12 + 0.46381060817777777779e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 19: {
      T t = 2 * y100 - 39;
      return 0.34104450552588334840e-1 +
          (0.10823541191350532574e-2 +
           (0.65354356159553934436e-5 +
            (0.37512918348533521149e-7 +
             (0.20362979635817883229e-9 +
              (0.10384187833037282363e-11 + 0.49300625262222222221e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 20: {
      T t = 2 * y100 - 41;
      return 0.36295603928292425716e-1 +
          (0.11089526167995268200e-2 +
           (0.67654845095518363577e-5 +
            (0.39184292949913591646e-7 +
             (0.21431552202133775150e-9 +
              (0.10994259106646731797e-11 + 0.52409949102222222221e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 21: {
      T t = 2 * y100 - 43;
      return 0.38540888038840509795e-1 +
          (0.11364917134175420009e-2 +
           (0.70058230641246312003e-5 +
            (0.40943644083718586939e-7 +
             (0.22563034723692881631e-9 +
              (0.11642841011361992885e-11 + 0.55721092871111111110e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 22: {
      T t = 2 * y100 - 45;
      return 0.40842225954785960651e-1 +
          (0.11650136437945673891e-2 +
           (0.72569945502343006619e-5 +
            (0.42796161861855042273e-7 +
             (0.23761401711005024162e-9 +
              (0.12332431172381557035e-11 + 0.59246802364444444445e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 23: {
      T t = 2 * y100 - 47;
      return 0.43201627431540222422e-1 +
          (0.11945628793917272199e-2 +
           (0.75195743532849206263e-5 +
            (0.44747364553960993492e-7 +
             (0.25030885216472953674e-9 +
              (0.13065684400300476484e-11 + 0.63000532853333333334e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 24: {
      T t = 2 * y100 - 49;
      return 0.45621193513810471438e-1 +
          (0.12251862608067529503e-2 +
           (0.77941720055551920319e-5 +
            (0.46803119830954460212e-7 +
             (0.26375990983978426273e-9 +
              (0.13845421370977119765e-11 + 0.66996477404444444445e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 25: {
      T t = 2 * y100 - 51;
      return 0.48103121413299865517e-1 +
          (0.12569331386432195113e-2 +
           (0.80814333496367673980e-5 +
            (0.48969667335682018324e-7 +
             (0.27801515481905748484e-9 +
              (0.14674637611609884208e-11 + 0.71249589351111111110e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 26: {
      T t = 2 * y100 - 53;
      return 0.50649709676983338501e-1 +
          (0.12898555233099055810e-2 +
           (0.83820428414568799654e-5 +
            (0.51253642652551838659e-7 +
             (0.29312563849675507232e-9 +
              (0.15556512782814827846e-11 + 0.75775607822222222221e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 27: {
      T t = 2 * y100 - 55;
      return 0.53263363664388864181e-1 +
          (0.13240082443256975769e-2 +
           (0.86967260015007658418e-5 +
            (0.53662102750396795566e-7 +
             (0.30914568786634796807e-9 +
              (0.16494420240828493176e-11 + 0.80591079644444444445e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 28: {
      T t = 2 * y100 - 57;
      return 0.55946601353500013794e-1 +
          (0.13594491197408190706e-2 +
           (0.90262520233016380987e-5 +
            (0.56202552975056695376e-7 +
             (0.32613310410503135996e-9 +
              (0.17491936862246367398e-11 + 0.85713381688888888890e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 29: {
      T t = 2 * y100 - 59;
      return 0.58702059496154081813e-1 +
          (0.13962391363223647892e-2 +
           (0.93714365487312784270e-5 +
            (0.58882975670265286526e-7 +
             (0.34414937110591753387e-9 +
              (0.18552853109751857859e-11 + 0.91160736711111111110e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 30: {
      T t = 2 * y100 - 61;
      return 0.61532500145144778048e-1 +
          (0.14344426411912015247e-2 +
           (0.97331446201016809696e-5 +
            (0.61711860507347175097e-7 +
             (0.36325987418295300221e-9 +
              (0.19681183310134518232e-11 + 0.96952238400000000000e-14 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 31: {
      T t = 2 * y100 - 63;
      return 0.64440817576653297993e-1 +
          (0.14741275456383131151e-2 +
           (0.10112293819576437838e-4 +
            (0.64698236605933246196e-7 +
             (0.38353412915303665586e-9 +
              (0.20881176114385120186e-11 + 0.10310784480000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 32: {
      T t = 2 * y100 - 65;
      return 0.67430045633130393282e-1 +
          (0.15153655418916540370e-2 +
           (0.10509857606888328667e-4 +
            (0.67851706529363332855e-7 +
             (0.40504602194811140006e-9 +
              (0.22157325110542534469e-11 + 0.10964842115555555556e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 33: {
      T t = 2 * y100 - 67;
      return 0.70503365513338850709e-1 +
          (0.15582323336495709827e-2 +
           (0.10926868866865231089e-4 +
            (0.71182482239613507542e-7 +
             (0.42787405890153386710e-9 +
              (0.23514379522274416437e-11 + 0.11659571751111111111e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 34: {
      T t = 2 * y100 - 69;
      return 0.73664114037944596353e-1 +
          (0.16028078812438820413e-2 +
           (0.11364423678778207991e-4 +
            (0.74701423097423182009e-7 +
             (0.45210162777476488324e-9 +
              (0.24957355004088569134e-11 + 0.12397238257777777778e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 35: {
      T t = 2 * y100 - 71;
      return 0.76915792420819562379e-1 +
          (0.16491766623447889354e-2 +
           (0.11823685320041302169e-4 +
            (0.78420075993781544386e-7 +
             (0.47781726956916478925e-9 +
              (0.26491544403815724749e-11 + 0.13180196462222222222e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 36: {
      T t = 2 * y100 - 73;
      return 0.80262075578094612819e-1 +
          (0.16974279491709504117e-2 +
           (0.12305888517309891674e-4 +
            (0.82350717698979042290e-7 +
             (0.50511496109857113929e-9 +
              (0.28122528497626897696e-11 + 0.14010889635555555556e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 37: {
      T t = 2 * y100 - 75;
      return 0.83706822008980357446e-1 +
          (0.17476561032212656962e-2 +
           (0.12812343958540763368e-4 +
            (0.86506399515036435592e-7 +
             (0.53409440823869467453e-9 +
              (0.29856186620887555043e-11 + 0.14891851591111111111e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 38: {
      T t = 2 * y100 - 77;
      return 0.87254084284461718231e-1 +
          (0.17999608886001962327e-2 +
           (0.13344443080089492218e-4 +
            (0.90900994316429008631e-7 +
             (0.56486134972616465316e-9 +
              (0.31698707080033956934e-11 + 0.15825697795555555556e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 39: {
      T t = 2 * y100 - 79;
      return 0.90908120182172748487e-1 +
          (0.18544478050657699758e-2 +
           (0.13903663143426120077e-4 +
            (0.95549246062549906177e-7 +
             (0.59752787125242054315e-9 +
              (0.33656597366099099413e-11 + 0.16815130613333333333e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 40: {
      T t = 2 * y100 - 81;
      return 0.94673404508075481121e-1 +
          (0.19112284419887303347e-2 +
           (0.14491572616545004930e-4 +
            (0.10046682186333613697e-6 +
             (0.63221272959791000515e-9 +
              (0.35736693975589130818e-11 + 0.17862931591111111111e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 41: {
      T t = 2 * y100 - 83;
      return 0.98554641648004456555e-1 +
          (0.19704208544725622126e-2 +
           (0.15109836875625443935e-4 +
            (0.10567036667675984067e-6 +
             (0.66904168640019354565e-9 +
              (0.37946171850824333014e-11 + 0.18971959040000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 42: {
      T t = 2 * y100 - 85;
      return 0.10255677889470089531e0 +
          (0.20321499629472857418e-2 +
           (0.15760224242962179564e-4 +
            (0.11117756071353507391e-6 +
             (0.70814785110097658502e-9 +
              (0.40292553276632563925e-11 + 0.20145143075555555556e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 43: {
      T t = 2 * y100 - 87;
      return 0.10668502059865093318e0 +
          (0.20965479776148731610e-2 +
           (0.16444612377624983565e-4 +
            (0.11700717962026152749e-6 +
             (0.74967203250938418991e-9 +
              (0.42783716186085922176e-11 + 0.21385479360000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 44: {
      T t = 2 * y100 - 89;
      return 0.11094484319386444474e0 +
          (0.21637548491908170841e-2 +
           (0.17164995035719657111e-4 +
            (0.12317915750735938089e-6 +
             (0.79376309831499633734e-9 +
              (0.45427901763106353914e-11 + 0.22696025653333333333e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 45: {
      T t = 2 * y100 - 91;
      return 0.11534201115268804714e0 +
          (0.22339187474546420375e-2 +
           (0.17923489217504226813e-4 +
            (0.12971465288245997681e-6 +
             (0.84057834180389073587e-9 +
              (0.48233721206418027227e-11 + 0.24079890062222222222e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 46: {
      T t = 2 * y100 - 93;
      return 0.11988259392684094740e0 +
          (0.23071965691918689601e-2 +
           (0.18722342718958935446e-4 +
            (0.13663611754337957520e-6 +
             (0.89028385488493287005e-9 +
              (0.51210161569225846701e-11 + 0.25540227111111111111e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 47: {
      T t = 2 * y100 - 95;
      return 0.12457298393509812907e0 +
          (0.23837544771809575380e-2 +
           (0.19563942105711612475e-4 +
            (0.14396736847739470782e-6 +
             (0.94305490646459247016e-9 +
              (0.54366590583134218096e-11 + 0.27080225920000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 48: {
      T t = 2 * y100 - 97;
      return 0.12941991566142438816e0 +
          (0.24637684719508859484e-2 +
           (0.20450821127475879816e-4 +
            (0.15173366280523906622e-6 +
             (0.99907632506389027739e-9 +
              (0.57712760311351625221e-11 + 0.28703099555555555556e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 49: {
      T t = 2 * y100 - 99;
      return 0.13443048593088696613e0 +
          (0.25474249981080823877e-2 +
           (0.21385669591362915223e-4 +
            (0.15996177579900443030e-6 +
             (0.10585428844575134013e-8 +
              (0.61258809536787882989e-11 + 0.30412080142222222222e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 50: {
      T t = 2 * y100 - 101;
      return 0.13961217543434561353e0 +
          (0.26349215871051761416e-2 +
           (0.22371342712572567744e-4 +
            (0.16868008199296822247e-6 +
             (0.11216596910444996246e-8 +
              (0.65015264753090890662e-11 + 0.32210394506666666666e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 51: {
      T t = 2 * y100 - 103;
      return 0.14497287157673800690e0 +
          (0.27264675383982439814e-2 +
           (0.23410870961050950197e-4 +
            (0.17791863939526376477e-6 +
             (0.11886425714330958106e-8 +
              (0.68993039665054288034e-11 + 0.34101266222222222221e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 52: {
      T t = 2 * y100 - 105;
      return 0.15052089272774618151e0 +
          (0.28222846410136238008e-2 +
           (0.24507470422713397006e-4 +
            (0.18770927679626136909e-6 +
             (0.12597184587583370712e-8 +
              (0.73203433049229821618e-11 + 0.36087889048888888890e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 53: {
      T t = 2 * y100 - 107;
      return 0.15626501395774612325e0 +
          (0.29226079376196624949e-2 +
           (0.25664553693768450545e-4 +
            (0.19808568415654461964e-6 +
             (0.13351257759815557897e-8 +
              (0.77658124891046760667e-11 + 0.38173420035555555555e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 54: {
      T t = 2 * y100 - 109;
      return 0.16221449434620737567e0 +
          (0.30276865332726475672e-2 +
           (0.26885741326534564336e-4 +
            (0.20908350604346384143e-6 +
             (0.14151148144240728728e-8 +
              (0.82369170665974313027e-11 + 0.40360957457777777779e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 55: {
      T t = 2 * y100 - 111;
      return 0.16837910595412130659e0 +
          (0.31377844510793082301e-2 +
           (0.28174873844911175026e-4 +
            (0.22074043807045782387e-6 +
             (0.14999481055996090039e-8 +
              (0.87348993661930809254e-11 + 0.42653528977777777779e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 56: {
      T t = 2 * y100 - 113;
      return 0.17476916455659369953e0 +
          (0.32531815370903068316e-2 +
           (0.29536024347344364074e-4 +
            (0.23309632627767074202e-6 +
             (0.15899007843582444846e-8 +
              (0.92610375235427359475e-11 + 0.45054073102222222221e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 57: {
      T t = 2 * y100 - 115;
      return 0.18139556223643701364e0 +
          (0.33741744168096996041e-2 +
           (0.30973511714709500836e-4 +
            (0.24619326937592290996e-6 +
             (0.16852609412267750744e-8 +
              (0.98166442942854895573e-11 + 0.47565418097777777779e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 58: {
      T t = 2 * y100 - 117;
      return 0.18826980194443664549e0 +
          (0.35010775057740317997e-2 +
           (0.32491914440014267480e-4 +
            (0.26007572375886319028e-6 +
             (0.17863299617388376116e-8 +
              (0.10403065638343878679e-10 + 0.50190265831111111110e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 59: {
      T t = 2 * y100 - 119;
      return 0.19540403413693967350e0 +
          (0.36342240767211326315e-2 +
           (0.34096085096200907289e-4 +
            (0.27479061117017637474e-6 +
             (0.18934228504790032826e-8 +
              (0.11021679075323598664e-10 + 0.52931171733333333334e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 60: {
      T t = 2 * y100 - 121;
      return 0.20281109560651886959e0 +
          (0.37739673859323597060e-2 +
           (0.35791165457592409054e-4 +
            (0.29038742889416172404e-6 +
             (0.20068685374849001770e-8 +
              (0.11673891799578381999e-10 + 0.55790523093333333334e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 61: {
      T t = 2 * y100 - 123;
      return 0.21050455062669334978e0 +
          (0.39206818613925652425e-2 +
           (0.37582602289680101704e-4 +
            (0.30691836231886877385e-6 +
             (0.21270101645763677824e-8 +
              (0.12361138551062899455e-10 + 0.58770520160000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 62: {
      T t = 2 * y100 - 125;
      return 0.21849873453703332479e0 +
          (0.40747643554689586041e-2 +
           (0.39476163820986711501e-4 +
            (0.32443839970139918836e-6 +
             (0.22542053491518680200e-8 +
              (0.13084879235290858490e-10 + 0.61873153262222222221e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 63: {
      T t = 2 * y100 - 127;
      return 0.22680879990043229327e0 +
          (0.42366354648628516935e-2 +
           (0.41477956909656896779e-4 +
            (0.34300544894502810002e-6 +
             (0.23888264229264067658e-8 +
              (0.13846596292818514601e-10 + 0.65100183751111111110e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 64: {
      T t = 2 * y100 - 129;
      return 0.23545076536988703937e0 +
          (0.44067409206365170888e-2 +
           (0.43594444916224700881e-4 +
            (0.36268045617760415178e-6 +
             (0.25312606430853202748e-8 +
              (0.14647791812837903061e-10 + 0.68453122631111111110e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 65: {
      T t = 2 * y100 - 131;
      return 0.24444156740777432838e0 +
          (0.45855530511605787178e-2 +
           (0.45832466292683085475e-4 +
            (0.38352752590033030472e-6 +
             (0.26819103733055603460e-8 +
              (0.15489984390884756993e-10 + 0.71933206364444444445e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 66: {
      T t = 2 * y100 - 133;
      return 0.25379911500634264643e0 +
          (0.47735723208650032167e-2 +
           (0.48199253896534185372e-4 +
            (0.40561404245564732314e-6 +
             (0.28411932320871165585e-8 +
              (0.16374705736458320149e-10 + 0.75541379822222222221e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 67: {
      T t = 2 * y100 - 135;
      return 0.26354234756393613032e0 +
          (0.49713289477083781266e-2 +
           (0.50702455036930367504e-4 +
            (0.42901079254268185722e-6 +
             (0.30095422058900481753e-8 +
              (0.17303497025347342498e-10 + 0.79278273368888888890e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 68: {
      T t = 2 * y100 - 137;
      return 0.27369129607732343398e0 +
          (0.51793846023052643767e-2 +
           (0.53350152258326602629e-4 +
            (0.45379208848865015485e-6 +
             (0.31874057245814381257e-8 +
              (0.18277905010245111046e-10 + 0.83144182364444444445e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 69: {
      T t = 2 * y100 - 139;
      return 0.28426714781640316172e0 +
          (0.53983341916695141966e-2 +
           (0.56150884865255810638e-4 +
            (0.48003589196494734238e-6 +
             (0.33752476967570796349e-8 +
              (0.19299477888083469086e-10 + 0.87139049137777777779e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 70: {
      T t = 2 * y100 - 141;
      return 0.29529231465348519920e0 +
          (0.56288077305420795663e-2 +
           (0.59113671189913307427e-4 +
            (0.50782393781744840482e-6 +
             (0.35735475025851713168e-8 +
              (0.20369760937017070382e-10 + 0.91262442613333333334e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 71: {
      T t = 2 * y100 - 143;
      return 0.30679050522528838613e0 +
          (0.58714723032745403331e-2 +
           (0.62248031602197686791e-4 +
            (0.53724185766200945789e-6 +
             (0.37827999418960232678e-8 +
              (0.21490291930444538307e-10 + 0.95513539182222222221e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 72: {
      T t = 2 * y100 - 145;
      return 0.31878680111173319425e0 +
          (0.61270341192339103514e-2 +
           (0.65564012259707640976e-4 +
            (0.56837930287837738996e-6 +
             (0.40035151353392378882e-8 +
              (0.22662596341239294792e-10 + 0.99891109760000000000e-13 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 73: {
      T t = 2 * y100 - 147;
      return 0.33130773722152622027e0 +
          (0.63962406646798080903e-2 +
           (0.69072209592942396666e-4 +
            (0.60133006661885941812e-6 +
             (0.42362183765883466691e-8 +
              (0.23888182347073698382e-10 + 0.10439349811555555556e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 74: {
      T t = 2 * y100 - 149;
      return 0.34438138658041336523e0 +
          (0.66798829540414007258e-2 +
           (0.72783795518603561144e-4 +
            (0.63619220443228800680e-6 +
             (0.44814499336514453364e-8 +
              (0.25168535651285475274e-10 + 0.10901861383111111111e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 75: {
      T t = 2 * y100 - 151;
      return 0.35803744972380175583e0 +
          (0.69787978834882685031e-2 +
           (0.76710543371454822497e-4 +
            (0.67306815308917386747e-6 +
             (0.47397647975845228205e-8 +
              (0.26505114141143050509e-10 + 0.11376390933333333333e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 76: {
      T t = 2 * y100 - 153;
      return 0.37230734890119724188e0 +
          (0.72938706896461381003e-2 +
           (0.80864854542670714092e-4 +
            (0.71206484718062688779e-6 +
             (0.50117323769745883805e-8 +
              (0.27899342394100074165e-10 + 0.11862637614222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 77: {
      T t = 2 * y100 - 155;
      return 0.38722432730555448223e0 +
          (0.76260375162549802745e-2 +
           (0.85259785810004603848e-4 +
            (0.75329383305171327677e-6 +
             (0.52979361368388119355e-8 +
              (0.29352606054164086709e-10 + 0.12360253370666666667e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 78: {
      T t = 2 * y100 - 157;
      return 0.40282355354616940667e0 +
          (0.79762880915029728079e-2 +
           (0.89909077342438246452e-4 +
            (0.79687137961956194579e-6 +
             (0.55989731807360403195e-8 +
              (0.30866246101464869050e-10 + 0.12868841946666666667e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 79: {
      T t = 2 * y100 - 159;
      return 0.41914223158913787649e0 +
          (0.83456685186950463538e-2 +
           (0.94827181359250161335e-4 +
            (0.84291858561783141014e-6 +
             (0.59154537751083485684e-8 +
              (0.32441553034347469291e-10 + 0.13387957943111111111e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 80: {
      T t = 2 * y100 - 161;
      return 0.43621971639463786896e0 +
          (0.87352841828289495773e-2 +
           (0.10002929142066799966e-3 +
            (0.89156148280219880024e-6 +
             (0.62480008150788597147e-8 +
              (0.34079760983458878910e-10 + 0.13917107176888888889e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 81: {
      T t = 2 * y100 - 163;
      return 0.45409763548534330981e0 +
          (0.91463027755548240654e-2 +
           (0.10553137232446167258e-3 +
            (0.94293113464638623798e-6 +
             (0.65972492312219959885e-8 +
              (0.35782041795476563662e-10 + 0.14455745872000000000e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 82: {
      T t = 2 * y100 - 165;
      return 0.47282001668512331468e0 +
          (0.95799574408860463394e-2 +
           (0.11135019058000067469e-3 +
            (0.99716373005509038080e-6 +
             (0.69638453369956970347e-8 +
              (0.37549499088161345850e-10 + 0.15003280712888888889e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 83: {
      T t = 2 * y100 - 167;
      return 0.49243342227179841649e0 +
          (0.10037550043909497071e-1 +
           (0.11750334542845234952e-3 +
            (0.10544006716188967172e-5 +
             (0.73484461168242224872e-8 +
              (0.39383162326435752965e-10 + 0.15559069118222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 84: {
      T t = 2 * y100 - 169;
      return 0.51298708979209258326e0 +
          (0.10520454564612427224e-1 +
           (0.12400930037494996655e-3 +
            (0.11147886579371265246e-5 +
             (0.77517184550568711454e-8 +
              (0.41283980931872622611e-10 + 0.16122419680000000000e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 85: {
      T t = 2 * y100 - 171;
      return 0.53453307979101369843e0 +
          (0.11030120618800726938e-1 +
           (0.13088741519572269581e-3 +
            (0.11784797595374515432e-5 +
             (0.81743383063044825400e-8 +
              (0.43252818449517081051e-10 + 0.16692592640000000000e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 86: {
      T t = 2 * y100 - 173;
      return 0.55712643071169299478e0 +
          (0.11568077107929735233e-1 +
           (0.13815797838036651289e-3 +
            (0.12456314879260904558e-5 +
             (0.86169898078969313597e-8 +
              (0.45290446811539652525e-10 + 0.17268801084444444444e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 87: {
      T t = 2 * y100 - 175;
      return 0.58082532122519320968e0 +
          (0.12135935999503877077e-1 +
           (0.14584223996665838559e-3 +
            (0.13164068573095710742e-5 +
             (0.90803643355106020163e-8 +
              (0.47397540713124619155e-10 + 0.17850211608888888889e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 88: {
      T t = 2 * y100 - 177;
      return 0.60569124025293375554e0 +
          (0.12735396239525550361e-1 +
           (0.15396244472258863344e-3 +
            (0.13909744385382818253e-5 +
             (0.95651595032306228245e-8 +
              (0.49574672127669041550e-10 + 0.18435945564444444444e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 89: {
      T t = 2 * y100 - 179;
      return 0.63178916494715716894e0 +
          (0.13368247798287030927e-1 +
           (0.16254186562762076141e-3 +
            (0.14695084048334056083e-5 +
             (0.10072078109604152350e-7 +
              (0.51822304995680707483e-10 + 0.19025081422222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 90: {
      T t = 2 * y100 - 181;
      return 0.65918774689725319200e0 +
          (0.14036375850601992063e-1 +
           (0.17160483760259706354e-3 +
            (0.15521885688723188371e-5 +
             (0.10601827031535280590e-7 +
              (0.54140790105837520499e-10 + 0.19616655146666666667e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 91: {
      T t = 2 * y100 - 183;
      return 0.68795950683174433822e0 +
          (0.14741765091365869084e-1 +
           (0.18117679143520433835e-3 +
            (0.16392004108230585213e-5 +
             (0.11155116068018043001e-7 +
              (0.56530360194925690374e-10 + 0.20209663662222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 92: {
      T t = 2 * y100 - 185;
      return 0.71818103808729967036e0 +
          (0.15486504187117112279e-1 +
           (0.19128428784550923217e-3 +
            (0.17307350969359975848e-5 +
             (0.11732656736113607751e-7 +
              (0.58991125287563833603e-10 + 0.20803065333333333333e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 93: {
      T t = 2 * y100 - 187;
      return 0.74993321911726254661e0 +
          (0.16272790364044783382e-1 +
           (0.20195505163377912645e-3 +
            (0.18269894883203346953e-5 +
             (0.12335161021630225535e-7 +
              (0.61523068312169087227e-10 + 0.21395783431111111111e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 94: {
      T t = 2 * y100 - 189;
      return 0.78330143531283492729e0 +
          (0.17102934132652429240e-1 +
           (0.21321800585063327041e-3 +
            (0.19281661395543913713e-5 +
             (0.12963340087354341574e-7 +
              (0.64126040998066348872e-10 + 0.21986708942222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 95: {
      T t = 2 * y100 - 191;
      return 0.81837581041023811832e0 +
          (0.17979364149044223802e-1 +
           (0.22510330592753129006e-3 +
            (0.20344732868018175389e-5 +
             (0.13617902941839949718e-7 +
              (0.66799760083972474642e-10 + 0.22574701262222222222e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 96: {
      T t = 2 * y100 - 193;
      return 0.85525144775685126237e0 +
          (0.18904632212547561026e-1 +
           (0.23764237370371255638e-3 +
            (0.21461248251306387979e-5 +
             (0.14299555071870523786e-7 +
              (0.69543803864694171934e-10 + 0.23158593688888888889e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 97: {
      T t = 2 * y100 - 195;
      return 0.89402868170849933734e0 +
          (0.19881418399127202569e-1 +
           (0.25086793128395995798e-3 +
            (0.22633402747585233180e-5 +
             (0.15008997042116532283e-7 +
              (0.72357609075043941261e-10 + 0.23737194737777777778e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 98: {
      T t = 2 * y100 - 197;
      return 0.93481333942870796363e0 +
          (0.20912536329780368893e-1 +
           (0.26481403465998477969e-3 +
            (0.23863447359754921676e-5 +
             (0.15746923065472184451e-7 +
              (0.75240468141720143653e-10 + 0.24309291271111111111e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
    case 99: {
      T t = 2 * y100 - 199;
      return 0.97771701335885035464e0 +
          (0.22000938572830479551e-1 +
           (0.27951610702682383001e-3 +
            (0.25153688325245314530e-5 +
             (0.16514019547822821453e-7 +
              (0.78191526829368231251e-10 + 0.24873652355555555556e-12 * t) *
                  t) *
                 t) *
                t) *
               t) *
          t;
    }
  }
  // we only get here if y = 1, i.e. |x| < 4*eps, in which case
  // erfcx is within 1e-15 of 1..
  return 1.0;
}

template <typename scalar_t>
static inline scalar_t calc_erfcx(scalar_t x) {
  if (at::_isnan(x)) {
    return x;
  }

  if (x >= 0) {
    if (x > 50) { // continued-fraction expansion is faster
      const scalar_t ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi)
      if (x > 5e7) { // 1-term expansion, important to avoid overflow
        return ispi / x;
      }
      /* 5-term expansion (rely on compiler for CSE), simplified from:
                ispi / (x+0.5/(x+1/(x+1.5/(x+2/x))))  */
      return ispi * ((x * x) * (x * x + 4.5) + 2) /
          (x * ((x * x) * (x * x + 5) + 3.75));
    }
    return erfcx_y100(400 / (4 + x));
  } else {
    if (x < -26.7) {
      return Numerics<scalar_t>::upper_bound();
    } else if (x < -6.1) {
      return 2 * Numerics<scalar_t>::exp(x * x);
    } else {
      return 2 * Numerics<scalar_t>::exp(x * x) - erfcx_y100(400 / (4 - x));
    }
  }
}

template <typename scalar_t>
static inline scalar_t calc_log_ndtr(scalar_t x) {
  scalar_t t = x * static_cast<scalar_t>(0.707106781186547524400844362104849);
  if (x < scalar_t{-1.0}) {
    return Numerics<scalar_t>::log(calc_erfcx(-t) / 2) - t * t;
  } else {
    return Numerics<scalar_t>::log1p(-Numerics<scalar_t>::erfc(t) / 2);
  }
}

template <typename scalar_t>
scalar_t chebyshev_polynomial_t_forward(scalar_t x, int64_t n) {
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(1.0)) {
    if (x > scalar_t(0.0) || n % 2 == 0) {
      return scalar_t(1.0);
    }

    return scalar_t(-1.0);
  }

  if ((n > 6) && (Numerics<scalar_t>::abs(x) < scalar_t(1.0))) {
    return Numerics<scalar_t>::cos(n * Numerics<scalar_t>::acos(x));
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x;
  scalar_t r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x) * q - p;
    p = q;
    q = r;
  }

  return r;
} // chebyshev_polynomial_t_forward(scalar_t  x, int64_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_t_forward(scalar_t x, scalar_t n) {
  return chebyshev_polynomial_t_forward(x, static_cast<int64_t>(n));
} // chebyshev_polynomial_t_forward(scalar_t  x, scalar_t  n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_u_forward(scalar_t x, int64_t n) {
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(1.0)) {
    if (x > scalar_t(0.0) || n % 2 == 0) {
      return n + 1;
    }

    return -(n + 1);
  }

  if ((n > 8) && (Numerics<scalar_t>::abs(x) < scalar_t(1.0))) {
    if (Numerics<scalar_t>::sin(Numerics<scalar_t>::acos(x)) != scalar_t(0.0)) {
      return Numerics<scalar_t>::sin((n + 1) * Numerics<scalar_t>::acos(x)) /
          Numerics<scalar_t>::sin(Numerics<scalar_t>::acos(x));
    }

    return (n + 1) *
        Numerics<scalar_t>::cos((n + 1) * Numerics<scalar_t>::acos(x)) / x;
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x + x;
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x + x;
  scalar_t r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x) * q - p;
    p = q;
    q = r;
  }

  return r;
} // chebyshev_polynomial_u_forward(scalar_t x, int64_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_u_forward(scalar_t x, scalar_t n) {
  return chebyshev_polynomial_u_forward(x, static_cast<int64_t>(n));
} // chebyshev_polynomial_u_forward(scalar_t x, scalar_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_v_forward(scalar_t x, int64_t n) {
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(1.0)) {
    if (x > scalar_t(0.0)) {
      return scalar_t(1.0);
    }

    if (n % 2 == 0) {
      return n + n + 1;
    }

    return -(n + n + 1);
  }

  if ((n > 8) && (Numerics<scalar_t>::abs(x) < scalar_t(1.0))) {
    if (Numerics<scalar_t>::sin(Numerics<scalar_t>::acos(x) / scalar_t(2.0)) !=
        scalar_t(1.0)) {
      return Numerics<scalar_t>::cos(
                 (n + scalar_t(0.5)) * Numerics<scalar_t>::acos(x)) /
          Numerics<scalar_t>::cos(Numerics<scalar_t>::acos(x) / scalar_t(2.0));
    }

    if (n % 2 == 0) {
      return n + n + 1;
    }

    return -(n + n + 1);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x + x - scalar_t(1.0);
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x + x - scalar_t(1.0);
  scalar_t r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x) * q - p;
    p = q;
    q = r;
  }

  return r;
} // chebyshev_polynomial_v_forward(scalar_t x, int64_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_v_forward(scalar_t x, scalar_t n) {
  return chebyshev_polynomial_v_forward(x, static_cast<int64_t>(n));
} // chebyshev_polynomial_v_forward(scalar_t x, scalar_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_w_forward(scalar_t x, int64_t n) {
  if (n < 0) {
    return scalar_t(0.0);
  }

  if (Numerics<scalar_t>::abs(x) == scalar_t(1.0)) {
    if (x > scalar_t(0.0)) {
      return n + n + 1;
    }

    if (n % 2 == 0) {
      return scalar_t(1.0);
    }

    return scalar_t(-1.0);
  }

  if ((n > 8) && (Numerics<scalar_t>::abs(x) < scalar_t(1.0))) {
    if (Numerics<scalar_t>::cos(Numerics<scalar_t>::acos(x) / scalar_t(2.0)) !=
        scalar_t(1.0)) {
      return Numerics<scalar_t>::sin(
                 (n + scalar_t(0.5)) * Numerics<scalar_t>::acos(x)) /
          Numerics<scalar_t>::sin(Numerics<scalar_t>::acos(x) / scalar_t(2.0));
    }

    if (x > scalar_t(0.0)) {
      return n + n + 1;
    }

    if (n % 2 == 0) {
      return scalar_t(1.0);
    }

    return scalar_t(-1.0);
  }

  if (n == 0) {
    return scalar_t(1.0);
  }

  if (n == 1) {
    return x + x + scalar_t(1.0);
  }

  scalar_t p = scalar_t(1.0);
  scalar_t q = x + x + scalar_t(1.0);
  scalar_t r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x) * q - p;
    p = q;
    q = r;
  }

  return r;
} // chebyshev_polynomial_w_forward(scalar_t x, int64_t n)

template <typename scalar_t>
scalar_t chebyshev_polynomial_w_forward(scalar_t x, scalar_t n) {
  return chebyshev_polynomial_w_forward(x, static_cast<int64_t>(n));
} // chebyshev_polynomial_w_forward(scalar_t x, scalar_t n)

template <typename T>
static inline T shifted_chebyshev_polynomial_t_forward(T x, int64_t n) {
  if (n < 0) {
    return T(0.0);
  }

  if (x == T(1.0)) {
    return T(1.0);
  }

  if (x == T(0.0)) {
    if (n % 2 == 0) {
      return T(1.0);
    }

    return T(-1.0);
  }

  if ((n > 6) && (std::abs(x + x - T(1.0)) < T(1.0))) {
    return std::cos(n * std::acos(x + x - T(1.0)));
  }

  if (n == 0) {
    return T(1.0);
  }

  if (n == 1) {
    return x + x - T(1.0);
  }

  T p = T(1.0);
  T q = x + x - T(1.0);
  T r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x - T(1.0) + (x + x - T(1.0))) * q - p;
    p = q;
    q = r;
  }

  return r;
} // shifted_chebyshev_polynomial_t_forward(T x, int64_t n)

template <typename T>
static inline T shifted_chebyshev_polynomial_t_forward(T x, T n) {
  return shifted_chebyshev_polynomial_t_forward(x, static_cast<int64_t>(n));
} // shifted_chebyshev_polynomial_t_forward(T x, T n)

template <typename T>
static inline C10_HOST_DEVICE T
shifted_chebyshev_polynomial_u_forward(T x, int64_t n) {
  if (n < 0) {
    return T(0.0);
  }

  if (x == T(1.0)) {
    return n + 1;
  }

  if (x == T(0.0)) {
    if (n % 2 == 0) {
      return n + 1;
    }

    return -(n + 1);
  }

  if ((n > 6) && (std::abs(x + x - T(1.0)) < T(1.0))) {
    if (std::sin(std::acos(x + x - T(1.0))) != T(0.0)) {
      return std::sin((n + 1) * std::acos(x + x - T(1.0))) /
          std::sin(std::acos(x + x - T(1.0)));
    }

    return (n + 1) * std::cos((n + 1) * std::acos(x + x - T(1.0))) /
        (x + x - T(1.0));
  }

  if (n == 0) {
    return T(1.0);
  }

  if (n == 1) {
    return x + x - T(1.0) + (x + x - T(1.0));
  }

  T p = T(1.0);
  T q = x + x - T(1.0) + (x + x - T(1.0));
  T r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x - T(1.0) + (x + x - T(1.0))) * q - p;
    p = q;
    q = r;
  }

  return r;
} // shifted_chebyshev_polynomial_u_forward(T x, int64_t n)

template <typename T>
static inline C10_HOST_DEVICE T
shifted_chebyshev_polynomial_u_forward(T x, T n) {
  return shifted_chebyshev_polynomial_u_forward(x, static_cast<int64_t>(n));
} // shifted_chebyshev_polynomial_u_forward(T x, T n)

template <typename T>
static inline C10_HOST_DEVICE T
shifted_chebyshev_polynomial_v_forward(T x, int64_t n) {
  if (n < 0) {
    return T(0.0);
  }

  if (x == T(1.0)) {
    return T(1.0);
  }

  if (x == T(0.0)) {
    if (n % 2 == 0) {
      return (n + n + 1);
    }

    return -(n + n + 1);
  }

  if ((n > 6) && (std::abs(x + x - T(1.0)) < T(1.0))) {
    if (std::sin(std::acos(x + x - T(1.0)) / T(2.0)) != T(1.0)) {
      return std::cos(((n) + T(0.5)) * std::acos(x + x - T(1.0))) /
          std::cos(std::acos(x + x - T(1.0)) / T(2.0));
    }

    if (n % 2 == 0) {
      return n + n + 1;
    }

    return -(n + n + 1);
  }

  if (n == 0) {
    return T(1.0);
  }

  if (n == 1) {
    return x + x - T(1.0) + (x + x - T(1.0)) - T(1.0);
  }

  T p = T(1.0);
  T q = x + x - T(1.0) + (x + x - T(1.0)) - T(1.0);
  T r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x - T(1.0) + (x + x - T(1.0))) * q - p;
    p = q;
    q = r;
  }

  return r;
} // shifted_chebyshev_polynomial_v_forward(T x, int64_t n)

template <typename T>
static inline C10_HOST_DEVICE T
shifted_chebyshev_polynomial_v_forward(T x, T n) {
  return shifted_chebyshev_polynomial_v_forward(x, static_cast<int64_t>(n));
} // shifted_chebyshev_polynomial_v_forward(T x, T n)

template <typename T>
static inline C10_HOST_DEVICE T
shifted_chebyshev_polynomial_w_forward(T x, int64_t n) {
  if (n < 0) {
    return T(0.0);
  }

  if (x == T(1.0)) {
    return n + n + 1;
  }

  if (x == T(0.0)) {
    if (n % 2 == 0) {
      return T(1.0);
    }

    return T(-1.0);
  }

  if ((n > 4) && (std::abs(x + x - T(1.0)) < T(1.0))) {
    if (std::cos(std::acos(x + x - T(1.0)) / T(2.0)) != T(1.0)) {
      return std::sin((n + T(0.5)) * std::acos(x + x - T(1.0))) /
          std::sin(std::acos(x + x - T(1.0)) / T(2.0));
    }

    if (n % 2 == 0) {
      return T(1.0);
    }

    return T(-1.0);
  }

  if (n == 0) {
    return T(1.0);
  }

  if (n == 1) {
    return x + x - T(1.0) + (x + x - T(1.0)) + T(1.0);
  }

  T p = T(1.0);
  T q = x + x - T(1.0) + (x + x - T(1.0)) + T(1.0);
  T r;

  for (int64_t k = 2; k <= n; k++) {
    r = (x + x - T(1.0) + (x + x - T(1.0))) * q - p;
    p = q;
    q = r;
  }

  return r;
} // shifted_chebyshev_polynomial_w_forward(T x, int64_t n)

template <typename T>
static inline C10_HOST_DEVICE T airy_ai_forward(T x) {
  static const T AN[] = {
      +3.46538101525629032477e-01f,
      +1.20075952739645805542e+01f,
      +7.62796053615234516538e+01f,
      +1.68089224934630576269e+02f,
      +1.59756391350164413639e+02f,
      +7.05360906840444183113e+01f,
      +1.40264691163389668864e+01f,
      +9.99999999999999995305e-01f,
  };

  static const T AD[] = {
      +5.67594532638770212846e-01f,
      +1.47562562584847203173e+01f,
      +8.45138970141474626562e+01f,
      +1.77318088145400459522e+02f,
      +1.64234692871529701831e+02f,
      +7.14778400825575695274e+01f,
      +1.40959135607834029598e+01f,
      +1.00000000000000000470e+00f,
  };

  static const T AFN[] = {
      -1.31696323418331795333e-01f,
      -6.26456544431912369773e-01f,
      -6.93158036036933542233e-01f,
      -2.79779981545119124951e-01f,
      -4.91900132609500318020e-02f,
      -4.06265923594885404393e-03f,
      -1.59276496239262096340e-04f,
      -2.77649108155232920844e-06f,
      -1.67787698489114633780e-08f,
  };

  static const T AFD[] = {
      +1.33560420706553243746e+01f,
      +3.26825032795224613948e+01f,
      +2.67367040941499554804e+01f,
      +9.18707402907259625840e+00f,
      +1.47529146771666414581e+00f,
      +1.15687173795188044134e-01f,
      +4.40291641615211203805e-03f,
      +7.54720348287414296618e-05f,
      +4.51850092970580378464e-07f,
  };

  static const T AGN[] = {
      +1.97339932091685679179e-02f,
      +3.91103029615688277255e-01f,
      +1.06579897599595591108e+00f,
      +9.39169229816650230044e-01f,
      +3.51465656105547619242e-01f,
      +6.33888919628925490927e-02f,
      +5.85804113048388458567e-03f,
      +2.82851600836737019778e-04f,
      +6.98793669997260967291e-06f,
      +8.11789239554389293311e-08f,
      +3.41551784765923618484e-10f,
  };

  static const T AGD[] = {
      +9.30892908077441974853e+00f,
      +1.98352928718312140417e+01f,
      +1.55646628932864612953e+01f,
      +5.47686069422975497931e+00f,
      +9.54293611618961883998e-01f,
      +8.64580826352392193095e-02f,
      +4.12656523824222607191e-03f,
      +1.01259085116509135510e-04f,
      +1.17166733214413521882e-06f,
      +4.91834570062930015649e-09f,
  };

  int domain_flag = 0;

  T ai;

  if (std::isinf(x)) {
    return std::numeric_limits<T>::quiet_NaN();
  }

  if (x > T(103.892f)) {
    return T(0.0f);
  }

  T f;
  T g;
  T k;

  if (x < T(-2.09f)) {
    T z = T(1.0f) / (T(-2.0f) * x * std::sqrt(-x) / T(3.0f));

    T afn = 0.0f;

    for (uint8_t index = 0; index <= 8; index++) {
      afn = afn * (z * z) + AFN[index];
    }

    T afd = 0.0f;

    for (uint8_t index = 0; index <= 8; index++) {
      afd = afd * (z * z) + AFD[index];
    }

    T agn = 0.0f;

    for (uint8_t index = 0; index <= 10 + 0; index++) {
      agn = agn * (z * z) + AGN[index];
    }

    T agd = 0.0f;

    for (uint8_t index = 0; index <= 10 - 1; index++) {
      agd = agd * (z * z) + AGD[index];
    }

    T t = T(-2.0f) * x * std::sqrt(-x) / T(3.0f) +
        T(0.25f) * T(3.14159265358979323846f);

    return T(5.64189583547756286948e-01f) / std::sqrt(std::sqrt(-x)) *
        (std::sin(t) * (T(1.0f) + z * z * afn / afd) -
         std::cos(t) * (z * agn / agd));
  }

  if (x >= T(2.09f)) {
    domain_flag = 5;

    T zeta = T(2.0f) * x * std::sqrt(x) / T(3.0f);

    T an = 0.0f;

    for (uint8_t index = 0; index <= 7; index++) {
      an = an * (T(1.0f) / zeta) + AN[index];
    }

    T ad = 0.0f;

    for (uint8_t index = 0; index <= 7; index++) {
      ad = ad * (T(1.0f) / zeta) + AD[index];
    }

    ai = T(5.64189583547756286948e-01f) * (an / ad) /
        (T(2.0f) * std::sqrt(std::sqrt(x)) * std::exp(zeta));

    if (x > T(8.3203353f)) {
      return ai;
    }
  }

  f = 1.0f;
  g = x;
  k = 1.0f;

  T m = 1.0f;
  T n = x;
  T t = 1.0f;
  T z = x * x * x;

  while (t > std::numeric_limits<T>::epsilon()) {
    m *= z;
    k += T(1.0f);
    m /= k;
    n *= z;
    k += T(1.0f);
    n /= k;
    m /= k;
    f += m;
    k += T(1.0f);
    n /= k;
    g += n;

    t = std::abs(m / f);
  }

  if ((domain_flag & 1) == 0) {
    return T(0.355028053887817239260f) * f - T(0.258819403792806798405f) * g;
  }

  return ai;
} // T airy_ai(T x)

} // namespace AtenIpexTypeXPU
} // namespace at
